Chapter 4.4, Problem 53E

Solution

To graph: The given function and determine its domain and range.

Domain: $\left(x>-1\right)$

Range: The range is all real numbers.

Given information: The function $h\left(x\right)={\log }_5\left(x+1\right)-3$

Concept used:

1. The graph of a function can be drawn by finding the points that lies on the curve of the given function.
2. The domain of an exponential function is the set of all real numbers.

Calculation:

To make the graph of the given function, substitute different values of $x$ in the given function and find the respective value of $y$ as follows: the graph for the parent function,

  $y={\log }_{5}x$

This can be represented by in exponential form,

  $\begin{array}{l}{\left(5\right)}^y=x\\ {\left(5\right)}^0=x\\ x=1\end{array}$

Similarly, find the values of $x$ at $y=0,1,2,$ and list them in the table given below:

  $\begin{array}{cccc}x& 1& 5& 25\\ y& 0& 1& 2\end{array}$

Now again for the parent graph left 1 units and down 3 units that passes through the point

  $\left(0,-3\right),\left(4,-2\right),\left(24,-1\right)$

Now, draw each point of form $\left(x,y\right)$ in a rectangular coordinate system and connect them with a smooth curve as follows:

  

From the graph, it is clear that the line $y=0$ is the asymptote of the given function; therefore, the domain of the given function is the entire set of real numbers and the range of the given function is set of all positive real numbers.

Note that the graph asymptote is $x=-1$ and the domain $\left(x>-1\right)$

The range is all real numbers.

Conclusion:

Hence, the domain of the given function is $\left(x>-1\right)$ and the range is all real numbers.