Question

find the equation of the exponential function.

$x$

$f(x)$

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0

0.1

1

$0.1 \cdot 3$

2

$0.1 \cdot (3)^2$

3

$0.1 \cdot (3)^3$

show your work herehint: to add an exponent ($x^y$), type \exponent\ or press \^\$f(x) =$

Explanation:

Step1: Recall exponential form

The standard 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 $x=0$, $f(0)=0.1$. Substitute into the form: $0.1 = a \cdot b^0$. Since $b^0=1$, $a=0.1$.

Step3: Identify growth factor $b$

Observe the pattern: for $x=1$, $f(1)=0.1 \cdot 3$; for $x=2$, $f(2)=0.1 \cdot 3^2$; for $x=3$, $f(3)=0.1 \cdot 3^3$. The base of the exponent is $3$, so $b=3$.

Step4: Write the final function

Substitute $a=0.1$ and $b=3$ into the standard form.

Answer:

$f(x) = 0.1 \cdot 3^x$