Chapter 0.3, Problem 45E

To determine

To State:

Which of the given options give the range of the given function.

Answer to Problem 45E

Option (b) $\left(-\infty ,4\right)$ .

Explanation of Solution

Given:

The function $y=4-2^{2x}$ .

The options:

  (a) $\left(-\infty ,\infty \right)$

  (b) $\left(-\infty ,4\right)$

  (c) $\left[-4,\infty \right)$

  (d) $\left(-\infty ,4\right]$

Concepts Used:

The exponential function $a^x;a>0$ has a range $\left(0,\infty \right)$ .

If the function $f\left(x\right)$ has a range $\left(a,b\right)$ then the function $-f\left(x\right)$ has the range $\left(-b,-a\right)$ .

If the function $f\left(x\right)$ has a range $\left(a,b\right)$ then the function $p+f\left(x\right)$ has the range $\left(p+a,p+b\right)$ . Here is a constant.

Calculations:

Consider the function $2^x$ . It is an exponential function with range $\left(0,\infty \right)$ . The negated function $-2^x$ will have the range $\left(-\infty ,0\right)$ . Further the shifted function $4-2^x$ will have the range $\left(4-\infty ,4+0\right)$ . Since, $4-\infty \to -\infty$ and $4+0=4$ , the range for the function $y=4-2^{2x}$ can be simplified to $\left(-\infty ,4\right)$ . Thus, $\left(-\infty ,4\right)$ is the range of the given function $y=4-2^{2x}$ .

Conclusion:

Only Option (b) $\left(-\infty ,4\right)$ matches the correct answer found above. None of the other options are equivalent to the correct answer.