કાર્ય અને સંબંધો (Functions and Relations)

અંકશાસ્ત્ર અને ગણિતમાં “કાર્ય” અને “સંબંધ” બંનેના ઉપયોગ સાથે ખૂબ જ મહત્વપૂર્ણ ગણિતીય ધોરણો છે.

1.1 કાર્યની વ્યાખ્યા (Definition of a Function)

કાર્ય એ એક ગણિતીય સિદ્ધાંત છે, જેમાં દરેક ઇનપુટ (ડોમેઇનના તત્વો) ને ચોક્કસપણે અનન્ય આઉટપુટ (રેન્જના તત્વો) સાથે જોડવામાં આવે છે. સાદું કહ્યું તો, દરેક ઇનપુટને એક અનન્ય આઉટપુટ મળે છે.

$y=f(x)$ અહીં $x$ ઇનપુટ છે અને $f(x)$ આઉટપુટ છે.

1.2 કાર્યના પ્રકારો (Types of Functions)

1.2.1 લિનિયર કાર્ય (Linear Function)

લિનિયર કાર્ય એ તે કાર્ય છે, જેમાં કાર્યના મૂલ્યો એક સીધી રેખા દ્વારા દર્શાવવામાં આવે છે.

$f(x)=mx+b$

અહીં $m$ ઢાળ છે અને $b$Intercept છે.

Example 1:

લિનિયર કાર્ય $f(x)=2x+3$ માટે વિવિધ મૂલ્યો શોધો.

Solution:

For $x=1$:

$f(x)=2(1)+3$

$=2+3$

$=5$

For $x=-2$:

$f(x)=2(-2)+3$

$=-4+3$

$=-1$

For $x=3$:

$f(x)=2(3)+3$

$=6+3$

$=9$

1.2.2 ક્વાડ્રેટીક કાર્ય (Quadratic Function)

ક્વાડ્રેટીક કાર્ય એ તે કાર્ય છે, જેમાં ઇનપુટ ત્રિજ્યા (square) દ્વારા દર્શાવેલું હોય છે.

$f(x)=a{x}^2+bx+c$

અહીં $a$, $b$, અને $c$ સમાંક છે.

Example 2:

ક્વાડ્રેટીક કાર્ય $f(x)=x^2–4x+3$ માટે ઇનપુટના મૂલ્યો શોધો.

Solution:

For $x=2$:

$f(x)=(2{)}^2–4(2)+3$

$=4–8+3$

$=-1$

For $x=-1$:

$f(x)=(-1{)}^2–4(-1)+3$

$=1+4+3$

$=8$

1.2.3 ક્યુબિક કાર્ય (Cubic Function)

ક્યુબિક કાર્ય એ તે છે, જ્યાં ઇનપુટનો ઘન (cube) કરવામાં આવે છે.

$f(x)=a{x}^3+b{x}^2+cx+d$

અહીં $a$, $b$, $c$, અને $d$ સ્થિર સમાંક છે.

Example 3:

ક્યુબિક કાર્ય $f(x)=x^3–2x^2+x+1$ માટે આઉટપુટ શોધો.

Solution:

For $x=1$:

$f(x)=(1{)}^3–2(1{)}^2+1+1$

$=1–2+1+1$

$=1$

For $x=-2$:

$f(x)=(-2{)}^3–2(-2{)}^2+(-2)+1$

$=-8–8–2+1$

$=-17$

1.2.4 અથંકાર્ય (Exponential Function)

અથંકાર્ય એ તે કાર્ય છે જ્યાં ઇનપુટનું સ્તરે (power) ઉત્પન્ન થાય છે.

$f(x)=a^x$

Example 4:

અથંકાર્ય $f(x)=2^x$ માટે ઇનપુટના મૂલ્યો શોધો.

Solution:

For $x=3$:

$f(x)=2^3$

$=8$

For $x=-1$:

$f(x)=2^{-1}$

$=\frac{1}{2}$

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1.3 સંબંધ (Relations)

સંબંધ એ ડોમેઇન અને રેન્જ વચ્ચેના જોડાણને દર્શાવે છે, જ્યાં દરેક ઇનપુટનો એક કે વધુ આઉટપુટ હોઈ શકે છે.

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1.4 Example Continuation for Functions and Relations

Example 5:

Evaluate the quadratic function $f(x)=3x^2–2x+1$ for given values of $x$.

Solution:

For $x=2$:

$f(x)=3(2{)}^2–2(2)+1$

$=3(4)–4+1$

$=12–4+1$

$=9$

For $x=-1$:

$f(x)=3(-1{)}^2–2(-1)+1$

$=3(1)+2+1$

$=6$

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Example 6:

Evaluate the cubic function $f(x)=x^3–3x^2+2x–5$.

Solution:

For $x=1$:

$f(x)=(1{)}^3–3(1{)}^2+2(1)–5$

$=1–3+2–5$

$=-5$

For $x=-2$:

$f(x)=(-2{)}^3–3(-2{)}^2+2(-2)–5$

$=-8–12–4–5$

$=-29$

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Example 7:

Evaluate the exponential function $f(x)=3^x$.

Solution:

For $x=4$:

$f(x)=3^4$

$=81$

For $x=-2$:

$f(x)=3^{-2}$

$=\frac{1}{3^2}$

$=\frac{1}{9}$

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Example 8:

Solve for $f(x)=2x+3$ and $g(x)=x^2–4$.

Find $f(2)$ and $g(3)$.

Solution:

For $f(2)$:

$f(x)=2(2)+3$

$=4+3$

$=7$

For $g(3)$:

$g(x)=(3{)}^2–4$

$=9–4$

$=5$

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Example 9:

Consider the linear function $f(x)=-4x+1$. Evaluate for $x=-2$ and $x=5$.

Solution:

For $x=-2$:

$f(x)=-4(-2)+1$

$=8+1$

$=9$

For $x=5$:

$f(x)=-4(5)+1$

$=-20+1$

$=-19$

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Example 10:

Evaluate the quadratic function $f(x)=x^2–4x+6$ for $x=3$ and $x=-1$.

Solution:

For $x=3$:

$f(x)=(3{)}^2–4(3)+6$

$=9–12+6$

$=3$

For $x=-1$:

$f(x)=(-1{)}^2–4(-1)+6$

$=1+4+6$

$=11$

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Example 11:

Solve for $f(x)=5x–7$ and evaluate at $x=4$ and $x=-3$.

Solution:

For $x=4$:

$f(x)=5(4)–7$

$=20–7$

$=13$

For $x=-3$:

$f(x)=5(-3)–7$

$=-15–7$

$=-22$

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Example 12:

Evaluate the function $f(x)=x^2+5x–8$ for $x=2$ and $x=-4$.

Solution:

For $x=2$:

$f(x)=(2{)}^2+5(2)–8$

$=4+10–8$

$=6$

For $x=-4$:

$f(x)=(-4{)}^2+5(-4)–8$

$=16–20–8$

$=-12$

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Example 13:

Evaluate the cubic function $f(x)=x^3+2x^2–4x+1$ for $x=1$ and $x=-2$.

Solution:

For $x=1$:

$f(x)=(1{)}^3+2(1{)}^2–4(1)+1$

$=1+2–4+1$

$=0$

For $x=-2$:

$f(x)=(-2{)}^3+2(-2{)}^2–4(-2)+1$

$=-8+8+8+1$

$=9$

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Example 14:

Evaluate the function $f(x)=3x^2–7x+2$ for $x=5$ and $x=-3$.

Solution:

For $x=5$:

$f(x)=3(5{)}^2–7(5)+2$

$=75–35+2$

$=42$

For $x=-3$:

$f(x)=3(-3{)}^2–7(-3)+2$

$=27+21+2$

$=50$

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Example 15:

Evaluate the quadratic function $f(x)=x^2+2x–5$ for $x=4$ and $x=-2$.

Solution:

For $x=4$:

$f(x)=(4{)}^2+2(4)–5$

$=16+8–5$

$=19$

For $x=-2$:

$f(x)=(-2{)}^2+2(-2)–5$

$=4–4–5$

$=-5$

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Example 16:

Evaluate the linear function $f(x)=-3x+7$ for $x=3$ and $x=-1$.

Solution:

For $x=3$:

$f(x)=-3(3)+7$

$=-9+7$

$=-2$

For $x=-1$:

$f(x)=-3(-1)+7$

$=3+7$

$=10$

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Example 17:

Evaluate the cubic function $f(x)=2x^3–x^2+4x–1$ for $x=2$ and $x=-3$.

Solution:

For $x=2$:

$f(x)=2(2{)}^3–(2{)}^2+4(2)–1$

$=2(8)–4+8–1$

$=16–4+8–1$

$=19$

For $x=-3$:

$f(x)=2(-3{)}^3–(-3{)}^2+4(-3)–1$

$=2(-27)–9–12–1$

$=-54–9–12–1$

$=-76$

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Example 18:

Solve for $f(x)=x^2–6x+9$ and evaluate for $x=5$ and $x=-2$.

Solution:

For $x=5$:

$f(x)=(5{)}^2–6(5)+9$

$=25–30+9$

$=4$

For $x=-2$:

$f(x)=(-2{)}^2–6(-2)+9$

$=4+12+9$

$=25$

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Example 19:

Evaluate the function $f(x)=3x^2+2x–1$ for $x=1$ and $x=-3$.

Solution:

For $x=1$:

$f(x)=3(1{)}^2+2(1)–1$

$=3+2–1$

$=4$

For $x=-3$:

$f(x)=3(-3{)}^2+2(-3)–1$

$=3(9)–6–1$

$=27–6–1$

$=20$

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Example 20:

Evaluate the quadratic function $f(x)=4x^2–5x+6$ for $x=2$ and $x=-4$.

Solution:

For $x=2$:

$f(x)=4(2{)}^2–5(2)+6$

$=4(4)–10+6$

$=16–10+6$

$=12$

For $x=-4$:

$f(x)=4(-4{)}^2–5(-4)+6$

$=4(16)+20+6$

$=64+20+6$

$=90$

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Example 21:

Evaluate the linear function $f(x)=-2x+4$ for $x=0$ and $x=5$.

Solution:

For $x=0$:

$f(x)=-2(0)+4$

$=0+4$

$=4$

For $x=5$:

$f(x)=-2(5)+4$

$=-10+4$

$=-6$

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Example 22:

Evaluate the cubic function $f(x)=x^3–2x^2+x–4$ for $x=3$ and $x=-1$.

Solution:

For $x=3$:

$f(x)=(3{)}^3–2(3{)}^2+3–4$

$=27–18+3–4$

$=8$

For $x=-1$:

$f(x)=(-1{)}^3–2(-1{)}^2+(-1)–4$

$=-1–2–1–4$

$=-8$

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Example 23:

Evaluate the quadratic function $f(x)=2x^2–4x+1$ for $x=4$ and $x=-2$.

Solution:

For $x=4$:

$f(x)=2(4{)}^2–4(4)+1$

$=2(16)–16+1$

$=32–16+1$

$=17$

For $x=-2$:

$f(x)=2(-2{)}^2–4(-2)+1$

$=2(4)+8+1$

$=8+8+1$

$=17$

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Example 24:

Evaluate the function $f(x)=5x^2–3x+2$ for $x=2$ and $x=-3$.

Solution:

For $x=2$:

$f(x)=5(2{)}^2–3(2)+2$

$=5(4)–6+2$

$=20–6+2$

$=16$

For $x=-3$:

$f(x)=5(-3{)}^2–3(-3)+2$

$=5(9)+9+2$

$=45+9+2$

$=56$

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Example 25:

Solve the linear function $f(x)=4x–7$ for $x=3$ and $x=-5$.

Solution:

For $x=3$:

$f(x)=4(3)–7$

$=12–7$

$=5$

For $x=-5$:

$f(x)=4(-5)–7$

$=-20–7$

$=-27$

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Example 26:

Evaluate the quadratic function $f(x)=x^2+3x–7$ for $x=5$ and $x=-4$.

Solution:

For $x=5$:

$f(x)=(5{)}^2+3(5)–7$

$=25+15–7$

$=33$

For $x=-4$:

$f(x)=(-4{)}^2+3(-4)–7$

$=16–12–7$

$=-3$

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Example 27:

Solve for $f(x)=6x–2$ and evaluate for $x=4$ and $x=-3$.

Solution:

For $x=4$:

$f(x)=6(4)–2$

$=24–2$

$=22$

For $x=-3$:

$f(x)=6(-3)–2$

$=-18–2$

$=-20$

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Example 28:

Evaluate the function $f(x)=2x^3–x^2+5x–1$ for $x=3$ and $x=-2$.

Solution:

For $x=3$:

$f(x)=2(3{)}^3–(3{)}^2+5(3)–1$

$=2(27)–9+15–1$

$=54–9+15–1$

$=59$

For $x=-2$:

$f(x)=2(-2{)}^3–(-2{)}^2+5(-2)–1$

$=2(-8)–4–10–1$

$=-16–4–10–1$

$=-31$

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Example 29:

Evaluate the quadratic function $f(x)=4x^2–6x+2$ for $x=2$ and $x=-3$.

Solution:

For $x=2$:

$f(x)=4(2{)}^2–6(2)+2$

$=4(4)–12+2$

$=16–12+2$

$=6$

For $x=-3$:

$f(x)=4(-3{)}^2–6(-3)+2$

$=4(9)+18+2$

$=36+18+2$

$=56$

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Example 30:

Evaluate the linear function $f(x)=-3x+5$ for $x=6$ and $x=-1$.

Solution:

For $x=6$:

$f(x)=-3(6)+5$

$=-18+5$

$=-13$

For $x=-1$:

$f(x)=-3(-1)+5$

$=3+5$

$=8$

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Example 31:

Solve for $f(x)=2x^2+3x–4$ and evaluate for $x=1$ and $x=-5$.

Solution:

For $x=1$:

$f(x)=2(1{)}^2+3(1)–4$

$=2+3–4$

$=1$

For $x=-5$:

$f(x)=2(-5{)}^2+3(-5)–4$

$=2(25)–15–4$

$=50–15–4$

$=31$

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Example 32:

Evaluate the cubic function $f(x)=x^3–4x+7$ for $x=4$ and $x=-2$.

Solution:

For $x=4$:

$f(x)=(4{)}^3–4(4)+7$

$=64–16+7$

$=55$

For $x=-2$:

$f(x)=(-2{)}^3–4(-2)+7$

$=-8+8+7$

$=7$

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Example 33:

Evaluate the quadratic function $f(x)=5x^2–3x+8$ for $x=2$ and $x=-4$.

Solution:

For $x=2$:

$f(x)=5(2{)}^2–3(2)+8$

$=5(4)–6+8$

$=20–6+8$

$=22$

For $x=-4$:

$f(x)=5(-4{)}^2–3(-4)+8$

$=5(16)+12+8$

$=80+12+8$

$=100$

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Example 34:

Evaluate the linear function $f(x)=-5x+9$ for $x=7$ and $x=-3$.

Solution:

For $x=7$:

$f(x)=-5(7)+9$

$=-35+9$

$=-26$

For $x=-3$:

$f(x)=-5(-3)+9$

$=15+9$

$=24$

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Example 35:

Solve for $f(x)=x^2+4x–6$ and evaluate for $x=5$ and $x=-4$.

Solution:

For $x=5$:

$f(x)=(5{)}^2+4(5)–6$

$=25+20–6$

$=39$

For $x=-4$:

$f(x)=(-4{)}^2+4(-4)–6$

$=16–16–6$

$=-6$

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Example 36:

Evaluate the function $f(x)=3x^3–2x^2+4x–9$ for $x=3$ and $x=-2$.

Solution:

For $x=3$:

$f(x)=3(3{)}^3–2(3{)}^2+4(3)–9$

$=3(27)–2(9)+12–9$

$=81–18+12–9$

$=66$

For $x=-2$:

$f(x)=3(-2{)}^3–2(-2{)}^2+4(-2)–9$

$=3(-8)–2(4)–8–9$

$=-24–8–8–9$

$=-49$

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Example 37:

Evaluate the quadratic function $f(x)=2x^2–3x+5$ for $x=1$ and $x=-6$.

Solution:

For $x=1$:

$f(x)=2(1{)}^2–3(1)+5$

$=2–3+5$

$=4$

For $x=-6$:

$f(x)=2(-6{)}^2–3(-6)+5$

$=2(36)+18+5$

$=72+18+5$

$=95$

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Example 38:

Solve for $f(x)=-2x+6$ and evaluate for $x=4$ and $x=-7$.

Solution:

For $x=4$:

$f(x)=-2(4)+6$

$=-8+6$

$=-2$

For $x=-7$:

$f(x)=-2(-7)+6$

$=14+6$

$=20$

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Example 39:

Evaluate the quadratic function $f(x)=x^2–2x+4$ for $x=3$ and $x=-5$.

Solution:

For $x=3$:

$f(x)=(3{)}^2–2(3)+4$

$=9–6+4$

$=7$

For $x=-5$:

$f(x)=(-5{)}^2–2(-5)+4$

$=25+10+4$

$=39$

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Example 40:

Evaluate the cubic function $f(x)=x^3–4x+6$ for $x=2$ and $x=-3$.

Solution:

For $x=2$:

$f(x)=(2{)}^3–4(2)+6$

$=8–8+6$

$=6$

For $x=-3$:

$f(x)=(-3{)}^3–4(-3)+6$

$=-27+12+6$

$=-9$

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Example 41:

Solve for $f(x)=3x^2+5x–2$ and evaluate for $x=4$ and $x=-1$.

Solution:

For $x=4$:

$f(x)=3(4{)}^2+5(4)–2$

$=3(16)+20–2$

$=48+20–2$

$=66$

For $x=-1$:

$f(x)=3(-1{)}^2+5(-1)–2$

$=3(1)–5–2$

$=3–5–2$

$=-4$

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Example 42:

Evaluate the linear function $f(x)=-7x+3$ for $x=6$ and $x=-2$.

Solution:

For $x=6$:

$f(x)=-7(6)+3$

$=-42+3$

$=-39$

For $x=-2$:

$f(x)=-7(-2)+3$

$=14+3$

$=17$

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Example 43:

Evaluate the quadratic function $f(x)=4x^2–6x+8$ for $x=2$ and $x=-4$.

Solution:

For $x=2$:

$f(x)=4(2{)}^2–6(2)+8$

$=4(4)–12+8$

$=16–12+8$

$=12$

For $x=-4$:

$f(x)=4(-4{)}^2–6(-4)+8$

$=4(16)+24+8$

$=64+24+8$

$=96$

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Example 44:

Evaluate the function $f(x)=x^2–3x+9$ for $x=1$ and $x=-5$.

Solution:

For $x=1$:

$f(x)=(1{)}^2–3(1)+9$

$=1–3+9$

$=7$

For $x=-5$:

$f(x)=(-5{)}^2–3(-5)+9$

$=25+15+9$

$=49$

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Example 45:

Evaluate the cubic function $f(x)=2x^3–x^2+5x+1$ for $x=2$ and $x=-1$.

Solution:

For $x=2$:

$f(x)=2(2{)}^3–(2{)}^2+5(2)+1$

$=2(8)–4+10+1$

$=16–4+10+1$

$=23$

For $x=-1$:

$f(x)=2(-1{)}^3–(-1{)}^2+5(-1)+1$

$=2(-1)–1–5+1$

$=-2–1–5+1$

$=-7$

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Example 46:

Evaluate the quadratic function $f(x)=5x^2–3x+2$ for $x=4$ and $x=-2$.

Solution:

For $x=4$:

$f(x)=5(4{)}^2–3(4)+2$

$=5(16)–12+2$

$=80–12+2$

$=70$

For $x=-2$:

$f(x)=5(-2{)}^2–3(-2)+2$

$=5(4)+6+2$

$=20+6+2$

$=28$

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Example 47:

Solve for $f(x)=x^2–4x+7$ and evaluate for $x=2$ and $x=-3$.

Solution:

For $x=2$:

$f(x)=(2{)}^2–4(2)+7$

$=4–8+7$

$=3$

For $x=-3$:

$f(x)=(-3{)}^2–4(-3)+7$

$=9+12+7$

$=28$

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Example 48:

Evaluate the linear function $f(x)=-2x+5$ for $x=4$ and $x=-6$.

Solution:

For $x=4$:

$f(x)=-2(4)+5$

$=-8+5$

$=-3$

For $x=-6$:

$f(x)=-2(-6)+5$

$=12+5$

$=17$

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Example 49:

Evaluate the quadratic function $f(x)=x^2+3x–4$ for $x=5$ and $x=-2$.

Solution:

For $x=5$:

$f(x)=(5{)}^2+3(5)–4$

$=25+15–4$

$=36$

For $x=-2$:

$f(x)=(-2{)}^2+3(-2)–4$

$=4–6–4$

$=-6$

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Example 50:

Evaluate the cubic function $f(x)=x^3+2x^2–5x+3$ for $x=3$ and $x=-1$.

Solution:

For $x=3$:

$f(x)=(3{)}^3+2(3{)}^2–5(3)+3$

$=27+18–15+3$

$=33$

For $x=-1$:

$f(x)=(-1{)}^3+2(-1{)}^2–5(-1)+3$

$=-1+2+5+3$

$=9$

---

Example 51:

Evaluate the quadratic function $f(x)=4x^2–3x+6$ for $x=3$ and $x=-4$.

Solution:

For $x=3$:

$f(x)=4(3{)}^2–3(3)+6$

$=4(9)–9+6$

$=36–9+6$

$=33$

For $x=-4$:

$f(x)=4(-4{)}^2–3(-4)+6$

$=4(16)+12+6$

$=64+12+6$

$=82$

---

Example 52:

Evaluate the cubic function $f(x)=x^3–2x^2+3x–4$ for $x=2$ and $x=-3$.

Solution:

For $x=2$:

$f(x)=(2{)}^3–2(2{)}^2+3(2)–4$

$=8–8+6–4$

$=2$

For $x=-3$:

$f(x)=(-3{)}^3–2(-3{)}^2+3(-3)–4$

$=-27–18–9–4$

$=-58$

---

Example 53:

Evaluate the quadratic function $f(x)=2x^2+5x–3$ for $x=4$ and $x=-1$.

Solution:

For $x=4$:

$f(x)=2(4{)}^2+5(4)–3$

$=2(16)+20–3$

$=32+20–3$

$=49$

For $x=-1$:

$f(x)=2(-1{)}^2+5(-1)–3$

$=2(1)–5–3$

$=2–5–3$

$=-6$

---

Example 54:

Evaluate the linear function $f(x)=-6x+7$ for $x=5$ and $x=-3$.

Solution:

For $x=5$:

$f(x)=-6(5)+7$

$=-30+7$

$=-23$

For $x=-3$:

$f(x)=-6(-3)+7$

$=18+7$

$=25$

---

Example 55:

Evaluate the quadratic function $f(x)=3x^2–4x+5$ for $x=2$ and $x=-2$.

Solution:

For $x=2$:

$f(x)=3(2{)}^2–4(2)+5$

$=3(4)–8+5$

$=12–8+5$

$=9$

For $x=-2$:

$f(x)=3(-2{)}^2–4(-2)+5$

$=3(4)+8+5$

$=12+8+5$

$=25$

---

Example 56:

Evaluate the cubic function $f(x)=2x^3+3x^2–4x+6$ for $x=2$ and $x=-1$.

Solution:

For $x=2$:

$f(x)=2(2{)}^3+3(2{)}^2–4(2)+6$

$=2(8)+3(4)–8+6$

$=16+12–8+6$

$=26$

For $x=-1$:

$f(x)=2(-1{)}^3+3(-1{)}^2–4(-1)+6$

$=2(-1)+3(1)+4+6$

$=-2+3+4+6$

$=11$

---

Example 57:

Solve for $f(x)=x^2–5x+7$ and evaluate for $x=3$ and $x=-2$.

Solution:

For $x=3$:

$f(x)=(3{)}^2–5(3)+7$

$=9–15+7$

$=1$

For $x=-2$:

$f(x)=(-2{)}^2–5(-2)+7$

$=4+10+7$

$=21$

---

Example 58:

Evaluate the linear function $f(x)=4x–6$ for $x=2$ and $x=-4$.

Solution:

For $x=2$:

$f(x)=4(2)–6$

$=8–6$

$=2$

For $x=-4$:

$f(x)=4(-4)–6$

$=-16–6$

$=-22$

---

Example 59:

Evaluate the quadratic function $f(x)=3x^2+4x–8$ for $x=5$ and $x=-1$.

Solution:

For $x=5$:

$f(x)=3(5{)}^2+4(5)–8$

$=3(25)+20–8$

$=75+20–8$

$=87$

For $x=-1$:

$f(x)=3(-1{)}^2+4(-1)–8$

$=3(1)–4–8$

$=3–4–8$

$=-9$

---

Example 60:

Evaluate the cubic function $f(x)=x^3–2x^2+4x–3$ for $x=3$ and $x=-2$.

Solution:

For $x=3$:

$f(x)=(3{)}^3–2(3{)}^2+4(3)–3$

$=27–18+12–3$

$=18$

For $x=-2$:

$f(x)=(-2{)}^3–2(-2{)}^2+4(-2)–3$

$=-8–8–8–3$

$=-27$

---

Example 61:

Evaluate the quadratic function $f(x)=5x^2–3x+9$ for $x=4$ and $x=-3$.

Solution:

For $x=4$:

$f(x)=5(4{)}^2–3(4)+9$

$=5(16)–12+9$

$=80–12+9$

$=77$

For $x=-3$:

$f(x)=5(-3{)}^2–3(-3)+9$

$=5(9)+9+9$

$=45+9+9$

$=63$

---

Example 62:

Solve for $f(x)=2x^2–4x+3$ and evaluate for $x=1$ and $x=-4$.

Solution:

For $x=1$:

$f(x)=2(1{)}^2–4(1)+3$

$=2–4+3$

$=1$

For $x=-4$:

$f(x)=2(-4{)}^2–4(-4)+3$

$=2(16)+16+3$

$=32+16+3$

$=51$

---

Example 63:

Evaluate the linear function $f(x)=7x–5$ for $x=6$ and $x=-2$.

Solution:

For $x=6$:

$f(x)=7(6)–5$

$=42–5$

$=37$

For $x=-2$:

$f(x)=7(-2)–5$

$=-14–5$

$=-19$

---

Example 64:

Evaluate the quadratic function $f(x)=4x^2–3x+2$ for $x=3$ and $x=-5$.

Solution:

For $x=3$:

$f(x)=4(3{)}^2–3(3)+2$

$=4(9)–9+2$

$=36–9+2$

$=29$

For $x=-5$:

$f(x)=4(-5{)}^2–3(-5)+2$

$=4(25)+15+2$

$=100+15+2$

$=117$

---

Example 65:

Solve for $f(x)=2x^2+5x–4$ and evaluate for $x=2$ and $x=-3$.

Solution:

For $x=2$:

$f(x)=2(2{)}^2+5(2)–4$

$=2(4)+10–4$

$=8+10–4$

$=14$

For $x=-3$:

$f(x)=2(-3{)}^2+5(-3)–4$

$=2(9)–15–4$

$=18–15–4$

$=-1$

---

Example 66:

Evaluate the function $f(x)=3x^3+2x^2–4x+5$ for $x=3$ and $x=-2$.

Solution:

For $x=3$:

$f(x)=3(3{)}^3+2(3{)}^2–4(3)+5$

$=3(27)+2(9)–12+5$

$=81+18–12+5$

$=92$

For $x=-2$:

$f(x)=3(-2{)}^3+2(-2{)}^2–4(-2)+5$

$=3(-8)+2(4)+8+5$

$=-24+8+8+5$

$=-3$

---

Example 67:

Evaluate the quadratic function $f(x)=x^2–6x+8$ for $x=2$ and $x=-4$.

Solution:

For $x=2$:

$f(x)=(2{)}^2–6(2)+8$

$=4–12+8$

$=0$

For $x=-4$:

$f(x)=(-4{)}^2–6(-4)+8$

$=16+24+8$

$=48$

---

Example 68:

Evaluate the quadratic function $f(x)=5x^2–2x+7$ for $x=4$ and $x=-3$.

Solution:

For $x=4$:

$f(x)=5(4{)}^2–2(4)+7$

$=5(16)–8+7$

$=80–8+7$

$=79$

For $x=-3$:

$f(x)=5(-3{)}^2–2(-3)+7$

$=5(9)+6+7$

$=45+6+7$

$=58$

---

Example 69:

Evaluate the cubic function $f(x)=x^3+2x^2–3x+4$ for $x=2$ and $x=-2$.

Solution:

For $x=2$:

$f(x)=(2{)}^3+2(2{)}^2–3(2)+4$

$=8+8–6+4$

$=14$

For $x=-2$:

$f(x)=(-2{)}^3+2(-2{)}^2–3(-2)+4$

$=-8+8+6+4$

$=10$

---

Example 70:

Solve for $f(x)=3x^2–7x+4$ and evaluate for $x=3$ and $x=-1$.

Solution:

For $x=3$:

$f(x)=3(3{)}^2–7(3)+4$

$=3(9)–21+4$

$=27–21+4$

$=10$

For $x=-1$:

$f(x)=3(-1{)}^2–7(-1)+4$

$=3(1)+7+4$

$=3+7+4$

$=14$

---

Example 71:

Evaluate the linear function $f(x)=-4x+5$ for $x=6$ and $x=-3$.

Solution:

For $x=6$:

$f(x)=-4(6)+5$

$=-24+5$

$=-19$

For $x=-3$:

$f(x)=-4(-3)+5$

$=12+5$

$=17$

---

Example 72:

Evaluate the quadratic function $f(x)=2x^2+3x–5$ for $x=2$ and $x=-4$.

Solution:

For $x=2$:

$f(x)=2(2{)}^2+3(2)–5$

$=2(4)+6–5$

$=8+6–5$

$=9$

For $x=-4$:

$f(x)=2(-4{)}^2+3(-4)–5$

$=2(16)–12–5$

$=32–12–5$

$=15$

---

Example 73:

Evaluate the cubic function $f(x)=x^3–x^2+4x–2$ for $x=3$ and $x=-2$.

Solution:

For $x=3$:

$f(x)=(3{)}^3–(3{)}^2+4(3)–2$

$=27–9+12–2$

$=28$

For $x=-2$:

$f(x)=(-2{)}^3–(-2{)}^2+4(-2)–2$

$=-8–4–8–2$

$=-22$

---

Example 74:

Solve for $f(x)=4x^2–5x+7$ and evaluate for $x=4$ and $x=-3$.

Solution:

For $x=4$:

$f(x)=4(4{)}^2–5(4)+7$

$=4(16)–20+7$

$=64–20+7$

$=51$

For $x=-3$:

$f(x)=4(-3{)}^2–5(-3)+7$

$=4(9)+15+7$

$=36+15+7$

$=58$

---

Example 75:

Evaluate the linear function $f(x)=3x–9$ for $x=5$ and $x=-2$.

Solution:

For $x=5$:

$f(x)=3(5)–9$

$=15–9$

$=6$

For $x=-2$:

$f(x)=3(-2)–9$

$=-6–9$

$=-15$

---

Example 76:

Evaluate the quadratic function $f(x)=6x^2–4x+5$ for $x=3$ and $x=-1$.

Solution:

For $x=3$:

$f(x)=6(3{)}^2–4(3)+5$

$=6(9)–12+5$

$=54–12+5$

$=47$

For $x=-1$:

$f(x)=6(-1{)}^2–4(-1)+5$

$=6(1)+4+5$

$=6+4+5$

$=15$

---

Example 77:

Evaluate the cubic function $f(x)=2x^3–3x^2+5x–7$ for $x=2$ and $x=-3$.

Solution:

For $x=2$:

$f(x)=2(2{)}^3–3(2{)}^2+5(2)–7$

$=2(8)–3(4)+10–7$

$=16–12+10–7$

$=7$

For $x=-3$:

$f(x)=2(-3{)}^3–3(-3{)}^2+5(-3)–7$

$=2(-27)–3(9)–15–7$

$=-54–27–15–7$

$=-103$

---

Example 78:

Solve for $f(x)=5x^2+2x–4$ and evaluate for $x=3$ and $x=-4$.

Solution:

For $x=3$:

$f(x)=5(3{)}^2+2(3)–4$

$=5(9)+6–4$

$=45+6–4$

$=47$

For $x=-4$:

$f(x)=5(-4{)}^2+2(-4)–4$

$=5(16)–8–4$

$=80–8–4$

$=68$

---

Example 79:

Evaluate the linear function $f(x)=-5x+6$ for $x=4$ and $x=-1$.

Solution:

For $x=4$:

$f(x)=-5(4)+6$

$=-20+6$

$=-14$

For $x=-1$:

$f(x)=-5(-1)+6$

$=5+6$

$=11$

---

Example 80:

Evaluate the quadratic function $f(x)=3x^2–2x+9$ for $x=1$ and $x=-2$.

Solution:

For $x=1$:

$f(x)=3(1{)}^2–2(1)+9$

$=3(1)–2+9$

$=3–2+9$

$=10$

For $x=-2$:

$f(x)=3(-2{)}^2–2(-2)+9$

$=3(4)+4+9$

$=12+4+9$

$=25$

---

Example 81:

Evaluate the cubic function $f(x)=x^3+x^2–4x+8$ for $x=2$ and $x=-3$.

Solution:

For $x=2$:

$f(x)=(2{)}^3+(2{)}^2–4(2)+8$

$=8+4–8+8$

$=12$

For $x=-3$:

$f(x)=(-3{)}^3+(-3{)}^2–4(-3)+8$

$=-27+9+12+8$

$=2$

---

Example 82:

Solve for $f(x)=4x^2+5x–6$ and evaluate for $x=3$ and $x=-2$.

Solution:

For $x=3$:

$f(x)=4(3{)}^2+5(3)–6$

$=4(9)+15–6$

$=36+15–6$

$=45$

For $x=-2$:

$f(x)=4(-2{)}^2+5(-2)–6$

$=4(4)–10–6$

$=16–10–6$

$=0$

---

Example 83:

Evaluate the linear function $f(x)=6x–4$ for $x=3$ and $x=-5$.

Solution:

For $x=3$:

$f(x)=6(3)–4$

$=18–4$

$=14$

For $x=-5$:

$f(x)=6(-5)–4$

$=-30–4$

$=-34$

---

Example 84:

Evaluate the quadratic function $f(x)=2x^2–4x+7$ for $x=5$ and $x=-1$.

Solution:

For $x=5$:

$f(x)=2(5{)}^2–4(5)+7$

$=2(25)–20+7$

$=50–20+7$

$=37$

For $x=-1$:

$f(x)=2(-1{)}^2–4(-1)+7$

$=2(1)+4+7$

$=2+4+7$

$=13$

---

Example 85:

Evaluate the cubic function $f(x)=x^3–x^2+6x–9$ for $x=4$ and $x=-3$.

Solution:

For $x=4$:

$f(x)=(4{)}^3–(4{)}^2+6(4)–9$

$=64–16+24–9$

$=63$

For $x=-3$:

$f(x)=(-3{)}^3–(-3{)}^2+6(-3)–9$

$=-27–9–18–9$

$=-63$

---

Example 86:

Solve for $f(x)=5x^2–6x+3$ and evaluate for $x=4$ and $x=-2$.

Solution:

For $x=4$:

$f(x)=5(4{)}^2–6(4)+3$

$=5(16)–24+3$

$=80–24+3$

$=59$

For $x=-2$:

$f(x)=5(-2{)}^2–6(-2)+3$

$=5(4)+12+3$

$=20+12+3$

$=35$

---

Example 87:

Evaluate the linear function $f(x)=-8x+7$ for $x=3$ and $x=-5$.

Solution:

For $x=3$:

$f(x)=-8(3)+7$

$=-24+7$

$=-17$

For $x=-5$:

$f(x)=-8(-5)+7$

$=40+7$

$=47$

---

Example 88:

Evaluate the quadratic function $f(x)=6x^2–2x+9$ for $x=1$ and $x=-3$.

Solution:

For $x=1$:

$f(x)=6(1{)}^2–2(1)+9$

$=6(1)–2+9$

$=6–2+9$

$=13$

For $x=-3$:

$f(x)=6(-3{)}^2–2(-3)+9$

$=6(9)+6+9$

$=54+6+9$

$=69$

---

Example 89:

Evaluate the cubic function $f(x)=3x^3+5x^2–7x+4$ for $x=2$ and $x=-1$.

Solution:

For $x=2$:

$f(x)=3(2{)}^3+5(2{)}^2–7(2)+4$

$=3(8)+5(4)–14+4$

$=24+20–14+4$

$=34$

For $x=-1$:

$f(x)=3(-1{)}^3+5(-1{)}^2–7(-1)+4$

$=3(-1)+5(1)+7+4$

$=-3+5+7+4$

$=13$

---

Example 90:

Solve for $f(x)=4x^2–3x+6$ and evaluate for $x=4$ and $x=-2$.

Solution:

For $x=4$:

$f(x)=4(4{)}^2–3(4)+6$

$=4(16)–12+6$

$=64–12+6$

$=58$

For $x=-2$:

$f(x)=4(-2{)}^2–3(-2)+6$

$=4(4)+6+6$

$=16+6+6$

$=28$

---

Example 91:

Evaluate the linear function $f(x)=5x–3$ for $x=7$ and $x=-4$.

Solution:

For $x=7$:

$f(x)=5(7)–3$

$=35–3$

$=32$

For $x=-4$:

$f(x)=5(-4)–3$

$=-20–3$

$=-23$

---

Example 92:

Evaluate the quadratic function $f(x)=7x^2–5x+3$ for $x=2$ and $x=-1$.

Solution:

For $x=2$:

$f(x)=7(2{)}^2–5(2)+3$

$=7(4)–10+3$

$=28–10+3$

$=21$

For $x=-1$:

$f(x)=7(-1{)}^2–5(-1)+3$

$=7(1)+5+3$

$=7+5+3$

$=15$

---

Example 93:

Evaluate the cubic function $f(x)=x^3–3x^2+4x–6$ for $x=3$ and $x=-2$.

Solution:

For $x=3$:

$f(x)=(3{)}^3–3(3{)}^2+4(3)–6$

$=27–27+12–6$

$=6$

For $x=-2$:

$f(x)=(-2{)}^3–3(-2{)}^2+4(-2)–6$

$=-8–12–8–6$

$=-34$

---

Example 94:

Solve for $f(x)=2x^2–5x+8$ and evaluate for $x=1$ and $x=-3$.

Solution:

For $x=1$:

$f(x)=2(1{)}^2–5(1)+8$

$=2(1)–5+8$

$=2–5+8$

$=5$

For $x=-3$:

$f(x)=2(-3{)}^2–5(-3)+8$

$=2(9)+15+8$

$=18+15+8$

$=41$

---

Example 95:

Evaluate the linear function $f(x)=-2x+9$ for $x=5$ and $x=-6$.

Solution:

For $x=5$:

$f(x)=-2(5)+9$

$=-10+9$

$=-1$

For $x=-6$:

$f(x)=-2(-6)+9$

$=12+9$

$=21$

---

Example 96:

Evaluate the quadratic function $f(x)=x^2+3x+5$ for $x=3$ and $x=-3$.

Solution:

For $x=3$:

$f(x)=(3{)}^2+3(3)+5$

$=9+9+5$

$=23$

For $x=-3$:

$f(x)=(-3{)}^2+3(-3)+5$

$=9–9+5$

$=5$

---

Example 97:

Evaluate the cubic function $f(x)=2x^3+4x^2–5x+7$ for $x=1$ and $x=-2$.

Solution:

For $x=1$:

$f(x)=2(1{)}^3+4(1{)}^2–5(1)+7$

$=2(1)+4(1)–5+7$

$=2+4–5+7$

$=8$

For $x=-2$:

$f(x)=2(-2{)}^3+4(-2{)}^2–5(-2)+7$

$=2(-8)+4(4)+10+7$

$=-16+16+10+7$

$=17$

---

Example 98:

Solve for $f(x)=3x^2–6x+8$ and evaluate for $x=2$ and $x=-3$.

Solution:

For $x=2$:

$f(x)=3(2{)}^2–6(2)+8$

$=3(4)–12+8$

$=12–12+8$

$=8$

For $x=-3$:

$f(x)=3(-3{)}^2–6(-3)+8$

$=3(9)+18+8$

$=27+18+8$

$=53$

---

Example 99:

Evaluate the linear function $f(x)=-3x+4$ for $x=7$ and $x=-4$.

Solution:

For $x=7$:

$f(x)=-3(7)+4$

$=-21+4$

$=-17$

For $x=-4$:

$f(x)=-3(-4)+4$

$=12+4$

$=16$

---

Example 100:

Evaluate the quadratic function $f(x)=5x^2–7x+10$ for $x=3$ and $x=-1$.

Solution:

For $x=3$:

$f(x)=5(3{)}^2–7(3)+10$

$=5(9)–21+10$

$=45–21+10$

$=34$

For $x=-1$:

$f(x)=5(-1{)}^2–7(-1)+10$

$=5(1)+7+10$

$=5+7+10$

$=22$