Question

solving quadratics & other equations - 12
17
x 0 ½ 1 3/2 2
y 4 8
18
x 0 ½ 1 3/2 2
y 5 15
go
topic: simplifying exponents
find the desired values for each function below.

1. f(x)=2x - 7

find f(-3)
find f(x)=21
find f(1/2)

1. g(x)=3^x(2)

find g(-4)
find g(x)=162
find g(1/2)

1. i(t)=210(1.08^t)

find i(12)
find i(t)=420
find i(1/2)

1. h(x)=x^2 + x - 6

find h(-5)
find h(x)=0
find h(1/2)

1. k(x)=-5x + 9

find k(-7)
find k(x)=0
find k(1/2)

1. m(x)=(5^x)2

find m(-2)
find m(x)=1
find m(1/2)
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Explanation:

Step1: Evaluate $f(-3)$ for $f(x)=2x - 7$

Substitute $x=-3$ into $f(x)$: $f(-3)=2\times(-3)-7=-6 - 7=-13$.

Step2: Solve $f(x)=21$ for $f(x)=2x - 7$

Set $2x - 7 = 21$. Add 7 to both sides: $2x=21 + 7=28$. Then divide by 2: $x = 14$.

Step3: Evaluate $f(\frac{1}{2})$ for $f(x)=2x - 7$

Substitute $x=\frac{1}{2}$ into $f(x)$: $f(\frac{1}{2})=2\times\frac{1}{2}-7=1 - 7=-6$.

Step4: Evaluate $g(-4)$ for $g(x)=3^{x}(2)$

Substitute $x = - 4$ into $g(x)$: $g(-4)=3^{-4}\times2=\frac{2}{3^{4}}=\frac{2}{81}$.

Step5: Solve $g(x)=162$ for $g(x)=3^{x}(2)$

Set $3^{x}\times2 = 162$. Divide both sides by 2: $3^{x}=81$. Since $3^{4}=81$, $x = 4$.

Step6: Evaluate $g(\frac{1}{2})$ for $g(x)=3^{x}(2)$

Substitute $x=\frac{1}{2}$ into $g(x)$: $g(\frac{1}{2})=3^{\frac{1}{2}}\times2 = 2\sqrt{3}$.

Step7: Evaluate $I(12)$ for $I(t)=210(1.08^{t})$

Substitute $t = 12$ into $I(t)$: $I(12)=210\times1.08^{12}\approx210\times2.51817\approx528.82$.

Step8: Solve $I(t)=420$ for $I(t)=210(1.08^{t})$

Set $210\times1.08^{t}=420$. Divide both sides by 210: $1.08^{t}=2$. Take the natural - logarithm of both sides: $t\ln(1.08)=\ln(2)$. Then $t=\frac{\ln(2)}{\ln(1.08)}\approx9.006$.

Step9: Evaluate $I(\frac{1}{2})$ for $I(t)=210(1.08^{t})$

Substitute $t=\frac{1}{2}$ into $I(t)$: $I(\frac{1}{2})=210\times1.08^{\frac{1}{2}}=210\sqrt{1.08}\approx210\times1.03923\approx218.24$.

Step10: Evaluate $h(-5)$ for $h(x)=x^{2}+x - 6$

Substitute $x=-5$ into $h(x)$: $h(-5)=(-5)^{2}+(-5)-6=25 - 5-6 = 14$.

Step11: Solve $h(x)=0$ for $h(x)=x^{2}+x - 6$

Set $x^{2}+x - 6=0$. Factor: $(x + 3)(x - 2)=0$. So $x=-3$ or $x = 2$.

Step12: Evaluate $h(\frac{1}{2})$ for $h(x)=x^{2}+x - 6$

Substitute $x=\frac{1}{2}$ into $h(x)$: $h(\frac{1}{2})=(\frac{1}{2})^{2}+\frac{1}{2}-6=\frac{1}{4}+\frac{1}{2}-6=\frac{1 + 2}{4}-6=\frac{3}{4}-6=-\frac{21}{4}$.

Step13: Evaluate $k(-7)$ for $k(x)=-5x + 9$

Substitute $x=-7$ into $k(x)$: $k(-7)=-5\times(-7)+9=35 + 9=44$.

Step14: Solve $k(x)=0$ for $k(x)=-5x + 9$

Set $-5x+9 = 0$. Subtract 9 from both sides: $-5x=-9$. Divide by - 5: $x=\frac{9}{5}$.

Step15: Evaluate $k(\frac{1}{2})$ for $k(x)=-5x + 9$

Substitute $x=\frac{1}{2}$ into $k(x)$: $k(\frac{1}{2})=-5\times\frac{1}{2}+9=-\frac{5}{2}+9=\frac{-5 + 18}{2}=\frac{13}{2}$.

Step16: Evaluate $m(-2)$ for $m(x)=(5^{x})2$

Substitute $x=-2$ into $m(x)$: $m(-2)=2\times5^{-2}=2\times\frac{1}{25}=\frac{2}{25}$.

Step17: Solve $m(x)=1$ for $m(x)=(5^{x})2$

Set $2\times5^{x}=1$. Divide both sides by 2: $5^{x}=\frac{1}{2}$. Take the natural - logarithm of both sides: $x\ln(5)=\ln(\frac{1}{2})=-\ln(2)$. Then $x=-\frac{\ln(2)}{\ln(5)}\approx - 0.4307$.

Step18: Evaluate $m(\frac{1}{2})$ for $m(x)=(5^{x})2$

Substitute $x=\frac{1}{2}$ into $m(x)$: $m(\frac{1}{2})=2\times5^{\frac{1}{2}}=2\sqrt{5}\approx4.472$.

Answer:

$f(-3)=-13$, $x = 14$ when $f(x)=21$, $f(\frac{1}{2})=-6$, $g(-4)=\frac{2}{81}$, $x = 4$ when $g(x)=162$, $g(\frac{1}{2})=2\sqrt{3}$, $I(12)\approx528.82$, $t\approx9.006$ when $I(t)=420$, $I(\frac{1}{2})\approx218.24$, $h(-5)=14$, $x=-3$ or $x = 2$ when $h(x)=0$, $h(\frac{1}{2})=-\frac{21}{4}$, $k(-7)=44$, $x=\frac{9}{5}$ when $k(x)=0$, $k(\frac{1}{2})=\frac{13}{2}$, $m(-2)=\frac{2}{25}$, $x\approx - 0.4307$ when $m(x)=1$, $m(\frac{1}{2})=2\sqrt{5}$