Chapter 6.4, Problem 53E

(a)

To determine

To find: whether the statement is always, sometimes, or never true.

Answer to Problem 53E

This statement is never true.

Explanation of Solution

Given information: The given function is $f\left(x\right)=\log x$.

As we know that an asymptote is a line that a curve approaches, but never touches.

  

Here, we can see that $x=0$ ,

Hence, a vertical translation of the graph $f\left(x\right)=\log x$ does not change the equation of the asymptote.

(b)

To determine

To find: whether the statement is always, sometimes, or never true.

Answer to Problem 53E

This statement is always true.

Explanation of Solution

Given information: The given function is $f\left(x\right)=e^x$.

As we know that an asymptote is a line that a curve approaches, but never touches.

  

Here, we can see that $y=0$ ,

Hence, a vertical translation of the graph $f\left(x\right)=e^x$ does change the equation of the asymptote.

(c)

To determine

To find: whether the statement is always, sometimes, or never true.

Answer to Problem 53E

This statement is always true.

Explanation of Solution

Given information: The given function is $f\left(x\right)=\log x$.

The graph of the given function is given below.

  

Here, we can see that domain is $\left(0,\infty \right),\left\{x|x>0\right\}$ ,

Hence, a horizontal shrink of the graph $f\left(x\right)=\log x$ does not change the equation of the domain.

(d)

To determine

To find: whether the statement is always, sometimes, or never true.

Answer to Problem 53E

This statement is sometimes true.

Explanation of Solution

Given information: The given function is $g\left(x\right)=a{b}^{k-h}+k$.

The value of variable decides whether the graph of $g\left(x\right)=a{b}^{k-h}+k$ intersects the x-axis or not. Hence, the given statement is sometimes true.