How to find the range of an exponential function?

Steps to Solve

1. Identify the general form of the exponential function

An exponential function typically has the form $f(x) = a \cdot b^x$, where $a$ is a constant, $b$ is the base, and $x$ is the exponent.

2. Determine the behavior of the base $b$

• If $0 < b < 1$, the function represents exponential decay.
• If $b > 1$, the function represents exponential growth.

3. Identify the range based on the constant $a$

The range of an exponential function $f(x) = a \cdot b^x$ without any transformations is determined by the sign and value of $a$:

• When $a > 0$, the function $f(x)$ is always greater than $0$ as $b^x > 0$ for all $x$.
• When $a < 0$, the function $f(x)$ is always less than $0$.

Therefore, if $a > 0$, the range is $(0, \infty)$, and if $a < 0$, the range is $(-\infty, 0)$.

4. Consider horizontal shifts

If the function includes a horizontal shift (of the form $f(x) = a \cdot b^{x-h}$), it does not affect the range of the function.

5. Consider vertical shifts

If the function includes a vertical shift (of the form $f(x) = a \cdot b^x + k$), it modifies the range:

• If $k$ is added or subtracted, it shifts the range up or down by $k$ units.

For example, if $f(x) = a \cdot b^x + k$, the range will be $(k, \infty)$ if $a > 0$, and $(-\infty, k)$ if $a < 0$.