Chapter 3.1, Problem 80E

To determine

To calculate: the domain, x-intercept, and vertical asymptote of the logarithmic function and sketch its graph.

Answer to Problem 80E

Domain of the given function is $\left(-\infty ,3\right)$

x-intercept of the given function is $x=2$

Vertical asymptote of the given function is $x=3$

Graph is

  

Explanation of Solution

Given: The given function is $f\left(x\right)=\ln \left(3-x\right)$

Concept used:

Domain: domain is the set of all possible inputs values that is x values for which the function is defined.

x- intercept: - if $y=f\left(x\right)$ be a function. Then the values of x for which y is 0 is called x- intercept

Vertical asymptote: Let $f\left(x\right)$ be a function, then the points at which the function is not defined are called vertical asymptote

Calculation:

As we know that logarithmic function is defined for positive values only.

So, $3-x>0\Rightarrow x<3$

So domain of the given function is $\left(-\infty ,3\right)$

To find x-intercept, keep $f\left(x\right)=0$ in given function, we get

  $\ln \left(3-x\right)=0$

Taking antilog both sides

  $\begin{array}{l}3-x=e^0\\ 3-x=1\\ x=2\end{array}$

So, x-intercept is 2

Now to find the vertical asymptote $f\left(x\right)=\ln \left(3-x\right)$ is undefined when $3-x=0$

  $\Rightarrow x=3$

To sketch the graph of $f\left(x\right)=\ln \left(3-x\right)$ ,

  $\begin{array}{ccccccc}x& -1& 0& 2& 2.5& 2.95& 2.99\\ y& 1.386& 1.099& 0& -0.693& -2.996& -4.605\end{array}$

Plot above values of x and y in graph and join then

  The final graph looks like