Chapter 7.6, Problem 70HP

a.

To determine

To identify if $g(x)=b^x+k$ has an asymptote of $y=k$ .

Answer to Problem 70HP

Yes

Explanation of Solution

Given:

  $y=g(x)=b^x+k$

Calculation:

Substitute $y=k$ in the given function.

  $\begin{array}{l}\Rightarrow k=b^x+k\\ \Rightarrow b^x=0\\ \Rightarrow x\to \infty \end{array}$

Hence, there is asymptote at $y=k$ , since the given function is exponential either it is monotonically increasing or decreasing graph, therefore there can never be any asymptote.

Conclusion:

Hence, the function $g(x)=b^x+k$ will never have an asymptote of $y=k$ .

b.

To determine

To identify if $j(x)=b^{x-h}$ passes through $(h,1)$

Answer to Problem 70HP

Yes

Explanation of Solution

Given:

  $y=j(x)=b^{x-h}$

Calculation:

Substitute $x=h$ in the given function.

  $\begin{array}{l}\Rightarrow y=b^{h-h}\\ \Rightarrow y=b^0\\ \Rightarrow y=1\end{array}$

Hence, there will always be a point $(h,1)$ from which the given function $y=j(x)=b^{x-h}$ will pass through, which means $(h,1)$ will always satisfy the given function.

Conclusion:

Hence, the function $j(x)=b^{x-h}$ will always pass through $(h,1)$ .

c.

To determine

To identify if $p(x)=-b^x$ always decreases as $x\to \infty$

Answer to Problem 70HP

No, The function $p(x)=-b^x$ always will not always decreases as $x\to \infty$ .

Explanation of Solution

Given:

  $y=p(x)=-b^x$

Case1: Considering b as positive.

For, $y=p(x)=-b^x$

The function will be monotonically decreasing as $x\to \infty$ .

Case2: Considering b as negative.

For, $y=p(x)=-b^x$

The function will be monotonically increasing as $x\to \infty$ .

Conclusion:

Hence, the given function will not always decrease as $x\to \infty$