Q. 52 - Precalculus Enhanced with Graphing Utilities

Question

Use transformations to graph the function. Determine the domain, range and horizontal asymptote of the function.

$f\left(x\right)=1-2^{-x/3}$

Step-by-Step Solution

Answer

The graph of the function is given below,

Domain: All real numbers.

Range: $\left\{y|y<1\right\} or \left(-\infty ,1\right)$

Horizontal asymptote: $y=1$

Step 1. Given information.

Consider the given function,

$f\left(x\right)=1-2^{-x/3}$

First, plot the graph of $y=2^x$, then replace x by $-x/3$ and reflect about the y-axis.

Again, multiply it by $-1$ and reflect about the x-axis.

Then add $1$ and shift up by $1unit$,

Step 2. Determine the characteristics of the function.

Consider the given function,

$f\left(x\right)=1-2^{-x/3}$

From the graph, the domain of the function is all real numbers. Then,

$x\to \infty ,f\left(x\to 1\right)$

The range is $\left(-\infty ,1\right)$.

Horizontal asymptote is $y=1$.