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Let
f be an exponential function of the form
f(x)=ab^(x). If
f(0)=4 and
f(3)=108, what is the equation of
f ?

Let $f$ be an exponential function of the form $f(x)=a b^{x}$ . If $f(0)=4$ and $f(3)=108$ , what is the equation of $f$ ?

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Linear approximation

Full solution

Q. Let

f

be an exponential function of the form

f(x)=a b^{x}

. If

f(0)=4

and

f(3)=108

, what is the equation of

f

?

Given Exponential Function: We are given an exponential function $f(x) = ab^{x}$ . To find the equation of $f$ , we need to determine the values of $a$ and $b$ . We are given two points on the function: $(0, 4)$ and $(3, 108)$ . We can use these points to create two equations.
Find Value of a: Using the point $(0, 4)$ , we substitute $x = 0$ and $f(x) = 4$ into the equation $f(x) = ab^{x}$ to find the value of a. $\newline$ $f(0) = ab^{0} = 4$ $\newline$ Since any number raised to the power of $0$ is $1$ , we have: $\newline$ $a \cdot 1 = 4$ $\newline$ Therefore, $a = 4$ .
Find Value of $b$ : Now we use the point $(3, 108)$ and our value for $a$ to find $b$ . We substitute $x = 3$ , $f(x) = 108$ , and $a = 4$ into the equation $f(x) = ab^{x}$ . $\newline$ $f(3) = 4b^{3} = 108$ $\newline$ To find $b$ , we divide both sides by $(3, 108)$ $0$ : $\newline$ $(3, 108)$ $1$ $\newline$ $(3, 108)$ $2$
Calculate Final Equation: To find the value of $b$ , we take the cube root of both sides of the equation $b^{3} = 27$ . $b = 27^{\frac{1}{3}}$ $b = 3$
Calculate Final Equation: To find the value of $b$ , we take the cube root of both sides of the equation $b^{3} = 27$ .
$b = 27^{\frac{1}{3}}$
$b = 3$ Now that we have both $a$ and $b$ , we can write the equation of the exponential function $f$ .
$f(x) = ab^{x}$
$f(x) = 4 \times 3^{x}$
This is the equation of the function $f$ .

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