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How does $h(x) = 8^x$ change over the interval from $x = 3$ to $x = 4$ ? $\newline$ Choices: $\newline$ (A) $h(x)$ increases by a factor of $8$ $\newline$ (B) $h(x)$ decreases by $8\%$ $\newline$ (C) $h(x)$ increases by $8$ $\newline$ (D) $h(x)$ increases by $8\%$

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Exponential functions over unit intervals

Full solution

Q. How does

h(x) = 8^x

change over the interval from

x = 3

to

x = 4

?

\newline

Choices:

\newline

(A)

h(x)

increases by a factor of

8

\newline

(B)

h(x)

decreases by

8\%

\newline

(C)

h(x)

increases by

8

\newline

(D)

h(x)

increases by

8\%

Calculate $h(3)$ : Calculate $h(3)$ by substituting $x = 3$ into $h(x) = 8^x$ . $\newline$ $h(3) = 8^3$ $\newline$ $h(3) = 512$
Calculate $h(4)$ : Now calculate $h(4)$ by substituting $x = 4$ into $h(x) = 8^x$ .
$h(4) = 8^4$
$h(4) = 4096$
Compare $h(3)$ and $h(4)$ : Compare $h(3)$ and $h(4)$ to determine the change. $\newline$ $h(3) = 512$ and $h(4) = 4096$ , so $h(x)$ increases.
Calculate factor increase: Calculate the factor by which $h(x)$ increases from $x = 3$ to $x = 4$ .
Factor increase = $\frac{h(4)}{h(3)}$
Factor increase = $\frac{4096}{512}$
Factor increase = $8$
Match calculated factor increase: Determine which choice matches the calculated factor increase. The correct choice is (A) $h(x)$ increases by a factor of $8$ .

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