Evaluate exponential functions

Evaluation of an exponential function means substituting a given input into a rule such as $f(x)=ab^x$ and determining the output, using exponent laws for zero, positive, and negative integer exponents and interpreting exact or calculator-based values. The quantities $a$ and $b$ represent the initial value and multiplicative growth or decay factor, respectively, supporting interpretation of tables, graphs, and real-world models; this scope does not include complex exponents, formal definitions for arbitrary real exponents, or calculus-based analysis.

Detailed Explanation: Evaluate exponential functions

To evaluate an exponential function, substitute the given input for $x$ and simplify using exponent rules.

For a function of the form

f(x)=ab^x,

$a$ is the initial value because $b^0=1$ .
$b$ is the growth or decay factor.
A negative exponent means take the reciprocal:

b^{-n}=\frac{1}{b^n}.

Example

Evaluate $f(-2)$ for

f(x)=200(0.8)^x.

Step 1: Substitute $-2$ for $x$ .

f(-2)=200(0.8)^{-2}

Step 2: Rewrite the negative exponent.

(0.8)^{-2}=\frac{1}{(0.8)^2}

So,

f(-2)=200\left(\frac{1}{(0.8)^2}\right)

Step 3: Calculate the power.

(0.8)^2=0.64

Therefore,

f(-2)=200\left(\frac{1}{0.64}\right) =\frac{200}{0.64} =312.5

Thus,

\boxed{f(-2)=312.5}

The output of the function when the input is $-2$ is $312.5$ .

Learn by doing: Evaluate exponential functions

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Exponential Function Solving - Growth (Discrete, Mis-matched Time Units) Equation to Value at Time

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