QuestionDetermine which statement is true for the function $f(x)=a\left(2^{x}\right)$ given the values of $a$ and $f(0)$ or $f(1)$ .

Studdy Solution

STEP 1

Assumptions1. The function given is $f(x)=a\left(^{x}\right)$ . We are trying to determine which of the three statements is true $f(0)=$ when $a=\frac{1}{}$ , $f(0)=3$ when $a=3$ , or $f(1)=9$ when $a=9$ .

STEP 2

The function $f(x)=a\left(2^{x}\right)$ is an exponential function. For any exponential function of the form $f(x)=ab^{x}$ , $f(0)$ is always equal to $a$ because any number raised to the power of0 is1.

STEP 3

We can use this property to check the first statement. If we plug in $x=0$ and $a=\frac{1}{2}$ into the function, we get $f(0)=\frac{1}{2}\left(2^{0}\right)$

STEP 4

implify the expression.
$f(0)=\frac{1}{2}\times1$

STEP 5

Calculate the value of $f(0)$ .
$f(0)=\frac{1}{2}$

STEP 6

The first statement, $f(0)=2$ when $a=\frac{1}{2}$ , is false because $f(0)=\frac{1}{2}$ when $a=\frac{1}{2}$ .

STEP 7

Next, we check the second statement. If we plug in $x=0$ and $a=3$ into the function, we get $f(0)=3\left(2^{0}\right)$

STEP 8

implify the expression.
$f(0)=3\times1$

STEP 9

Calculate the value of $f()$ .
$f()=3$

STEP 10

The second statement, $f(0)=3$ when $a=3$ , is true because $f(0)=3$ when $a=3$ .

STEP 11

Finally, we check the third statement. If we plug in $x=$ and $a=9$ into the function, we get $f()=9\left(^{}\right)$

STEP 12

implify the expression.
$f()=9\times2$

STEP 13

Calculate the value of $f()$ .
$f()=18$

STEP 14

The third statement, $f()=9$ when $a=9$ , is false because $f()=18$ when $a=9$ .
So, the only true statement is $f(0)=3$ when $a=3$ .

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