Chapter 1, Problem 35RE

(a)

To determine

To find: The domain of the function $y=\ln \left(x-3\right)+1$ .

Answer to Problem 35RE

The domain of the function $y=\ln \left(x-3\right)+1$ is the interval $\left(3,\infty \right)$ .

Explanation of Solution

Given information:

The given function is $y=\ln \left(x-3\right)+1$ .

Calculation:

The function $f\left(x\right)=\ln \left(x-3\right)$ is defined only if $x-3>0$ i.e. $x>3$ .

So, the function $y=\ln \left(x-3\right)+1$ is defined for all the real values of $x>3$ .

Therefore, the domain of the function $y=\ln \left(x-3\right)+1$ is the interval $\left(3,\infty \right)$ .

(b)

To determine

To find: The range of the function $y=\ln \left(x-3\right)+1$ .

Answer to Problem 35RE

The range of the function $y=\ln \left(x-3\right)+1$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given information:

The given function is $y=\ln \left(x-3\right)+1$ .

Calculation:

The value of the function $f\left(x\right)=\ln x$ is the set of real numbers.

So, the value of the function $y=\ln \left(x-3\right)+1$ is the set of real numbers.

Therefore, the range of the function $y=\ln \left(x-3\right)+1$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

(c)

To determine

To graph: Thefunction $y=\ln \left(x-3\right)+1$ .

Explanation of Solution

Given information:

The given functionis $y=\ln \left(x-3\right)+1$ .

Graph:

To graph a function $y=\ln \left(x-3\right)+1$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\ln \left(x-3\right)+1$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{LN}}\boxed{\text{X}}\boxed{-}\boxed{3}\boxed{)}\boxed{+}\boxed{1}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\ln \left(\text{X}-3\right)+1$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of both the equations as shown below.

  

Figure (1)

Interpretation: From the above graph it can be observed that the graph the function $y=\ln \left(x-3\right)+1$ approaches to infinity as the values of x approaches to infinity.