Chapter 1.5, Problem 61E

(a)

To determine

To find:The domain and range of the function $y=a\left(b^{c-x}\right)+d$ .

Answer to Problem 61E

The domain of the function $y=a\left(b^{c-x}\right)+d$ is $\left(-\infty ,\infty \right)$ , and range is $\left(d,\infty \right)$ if $a>0$ and $\left(-\infty ,d\right)$ if $a<0$ .

Explanation of Solution

Given information:The function is $y=a\left(b^{c-x}\right)+d$ , and suppose that $a\ne 0$ , $b\ne 1$ , and $b>0$ .

Calculation:

The given function is an exponential function, and so the exponent $c-x$ can take any real value.

If the exponent $c-x$ can be any real number, then $x$ be any real number. So, the domain of the function is $\left(-\infty ,\infty \right)$ .

An exponent with a positive base will always return a positive number. Thus, it is known that $b^{c-x}>0$ .

Multiply both sides of the inequality $b^{c-x}>0$ by $a$ , and there are two possibility to solve this inequality.

If $a>0$ , then

  $\begin{array}{c}a{b}^{c-x}>0\\ a{b}^{c-x}+d>d\end{array}$

For this case, the range will be $\left(d,\infty \right)$ .

If $a<0$ , then

  $\begin{array}{c}a{b}^{c-x}<0\\ a{b}^{c-x}+d<d\end{array}$

For this case, the range will be $\left(-\infty ,d\right)$ .

Therefore, the domain of the function $y=a\left(b^{c-x}\right)+d$ is $\left(-\infty ,\infty \right)$ , and range is $\left(d,\infty \right)$ if $a>0$ and $\left(-\infty ,d\right)$ if $a<0$ .

(b)

To determine

To find:The domain and range of the function $y=a{\log }_b\left(x-c\right)+d$ .

Answer to Problem 61E

The domain and range of the function $y=a{\log }_b\left(x-c\right)+d$ are $\left(c,\infty \right)$ , and $\left(-\infty ,\infty \right)$ respectively.

Explanation of Solution

Given information:The function is $y=a{\log }_b\left(x-c\right)+d$ , and suppose that $a\ne 0$ , $b\ne 1$ , and $b>0$ .

Calculation:

The given function is a logarithmic function, and so the argument of the logarithm must be positive.

  $\begin{array}{c}x-c>0\\ x>c\end{array}$

So, the domain of the function will be $\left(c,\infty \right)$ .

A logarithmic function will always have a range of all real numbers regardless of transformations. So, the range will be $\left(-\infty ,\infty \right)$ ,

Therefore, the domain and range of the function $y=a{\log }_b\left(x-c\right)+d$ are $\left(c,\infty \right)$ , and $\left(-\infty ,\infty \right)$ respectively.