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Chapter 0: Problem 69

Determine whether the statement is true or false. Justify your answer. The graphs of \(f(x)=|x|+6\) and \(f(x)=|-x|+6\) are identical.

Short Answer

Expert verified
The statement is true. The graphs of the given functions \(f(x)=|x|+6\) and \(f(x)=|-x|+6\) are identical. Both functions represent the same mathematical relationship, which means their graphs are identical.

Step by step solution

01

Interpret the functions

The first function is \(f(x)=|x|+6\) which is a translation 6 units up from the function \(f(x)=|x|\) along y-axis. The second function is \(f(x)=|-x|+6\), which is the absolute value of the negative of \(x\), then shifted 6 units up along y-axis.
02

Analyze the absolute values

In general, the absolute value of a number is always positive or zero, thus \(|x|\) and \(|-x|\) will have the same values. That is, whether you take the absolute value of \(x\) or \(-x\), the result is always the same.
03

Compare graphically

Both these functions graph as a 'V' shape, which meets at the point (0,6), so the graphs of both functions overlap each other. Hence, both the graphs are identical.
04

Compare mathematically

Mathematically comparing both the functions, since the absolute value of \(x\) and \(-x\) are equal for all defined values of \(x\), both the functions have the same output for all given input, meaning they are identical.

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Most popular questions from this chapter

In Exercises 69-74, use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ \left(f^{-1} \circ f^{-1}\right)(6) $$

Use a graphing utility to graph each function. Write a paragraph describing any similarities and differences you observe among the graphs. (a) \(y=x\) (b) \(y=x^{2}\) (c) \(y=x^{3}\) (d) \(y=x^{4}\) (e) \(y=x^{5}\) (f) \(y=x^{6}\)

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\sqrt[3]{x-1} $$

True or False? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. Proof Prove that if \(f\) is a one-to-one odd function, then \(f^{-1}\) is an odd function.

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