Question

The graph of the exponential function f(x)=3^x+3 is given with three points. Determine the following for the graph of f^-1(x).1. Graph f^-1(x)2. Find the domain of f^-1(x)3. Find the range of f^-1(x)4. Does f^-1(x) increase or decrease on its domain?5. The equation of the vertical asymptote for f^-1(x) is?

Answer 1

Let's begin by listing out the given information:

`\begin{gathered} f\mleft(x\mright)=3^x+3\Rightarrow y=3^x+3 \\ Switch\text{ }the\text{ }variables\text{ x \& y in the equation, we have:} \\ x=3^y+3\Rightarrow x-3=3^y \\ x-3=3^y\Rightarrow3^y=x-3 \\ \text{Take the }ln\text{ }o\text{f both }sides\text{, we have:} \\ y=(ln(\left(x - 3 \right)))/(ln(\left(3 \right))) \\ y=f^(-1)\mleft(x\mright) \\ f^(-1)\mleft(x\mright)=(ln((x-3)))/(ln((3))) \\ \end{gathered}`

2.

The domain of a function is the set of input or argument values for which the function is real and defined. This is given by:

`\begin{gathered} x-3;x>3 \\ \therefore x>3 \end{gathered}`

3.

The range of a function is the set of output values for which the function is defined. This is given by:

Therefore, the domain of f^-1(x) increases

5.

The asymptote of a curve is a line such that the distance between the curve and the line approaches zero

`\begin{gathered} \: (\ln\left(x-3\right))/(\ln\left(3\right))\colon\quad \mathrm{Vertical}\colon\: x=3 \\ \therefore x=3 \end{gathered}`