Question

find the equation of the exponential function.

x	f(x)
1	4·2
2	4·(2)²
3	4·(2)³

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Explanation:

Step1: Recall exponential form

The general form of an exponential function is $f(x) = ab^x$, where $a$ is the initial value (when $x=0$), and $b$ is the base.

Step2: Identify $a$ from $x=0$

When $x=0$, $f(0)=4$. Substitute into the general form: $4 = ab^0$. Since $b^0=1$, we get $a=4$.

Step3: Identify base $b$

Look at the pattern of $f(x)$ values:

• When $x=1$, $f(1)=4\cdot2 = 4\cdot2^1$
• When $x=2$, $f(2)=4\cdot2^2$
• When $x=3$, $f(3)=4\cdot2^3$

This shows $b=2$.

Step4: Write the function

Substitute $a=4$ and $b=2$ into the general exponential form.

Answer:

$f(x)=4\cdot2^x$