Question

find the equation of the exponential function.
$x$ $f(x)$
0 1
1 $1\cdot2$
2 $1\cdot(2)^2$
3 $1\cdot(2)^3$
show your work here
hint: to add an exponent ($x^b$), type \exponent\ or press \^\

Explanation:

Step1: Recall exponential form

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

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

When $x=0$, $f(0)=1$. Substitute into the formula:
$1 = a \cdot b^0$
Since $b^0=1$ for any $b
eq 0$, we get $a=1$.

Step3: Identify growth factor $b$

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

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

This shows the base $b=2$.

Step4: Write the final function

Substitute $a=1$ and $b=2$ into the general form.

Answer:

$f(x) = 2^x$ (or $f(x)=1\cdot2^x$)