Questions: Graph the exponential function g(x) = 2^x - 2. To do this, plot two points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button. Additionally, give the domain and range of the function using interval notation. Domain: Range:

Graph the exponential function g(x) = 2^x - 2.
To do this, plot two points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button.
Additionally, give the domain and range of the function using interval notation.

Domain:

Range:

Transcript text: Graph the exponential function $g(x)=2^{x}-2$. To do this, plot two points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button. Additionally, give the domain and range of the function using interval notation. Domain: $\square$ Range: $\square$

Solution

Solution Steps

Step 1: Identify the Parameters

The given exponential function is $g(x)=1\cdot 2^{x-0}-2$. Here, $a=1$, $b=2$, $c=0$, and $d=-2$.

Step 2: Determine the Asymptote

The horizontal asymptote of the function is the line $y=-2$. This is because as $x$ approaches infinity or negative infinity, the value of $2^{x-0}$ approaches 0 if $0<2<1$ or grows infinitely if $2>1$, but the function value approaches $-2$.

Step 3: Plot Two Points

Choosing $x=0$ simplifies the exponent, giving us one point: ($0$, $-1$). Choosing another $x$ value slightly greater than $c$, say $x=1$, gives us another point: ($1$, $0$).

Step 4: Draw the Graph

Since $a>0$ and $b>1$, the graph increases. Reflect or stretch/compress the graph as indicated by the value of $a$.

Step 5: Domain and Range

The domain of the function is all real numbers, $(-\infty, \infty)$, because you can input any real number into an exponential function. The range of the function is $(-2, \infty)$.