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Chapter 1: Problem 73

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
True. The graphs of \(f(x)=|x|+6\) and \(f(x)=|-x|+6\) are identical.

Step by step solution

01

Understanding Absolute Value

An absolute value function converts any negative input into a positive output, but leaves positive input unchanged. So both \(f(x)=|x|+6\) and \(f(x)=|-x|+6\) will always produce positive outputs or zero, never negatives.
02

Consideration of the Two Functions

Looking at both functions, it can be seen that the only difference is the sign inside the absolute value symbols. For \(f(x)=|x|+6\), if \(x\) is negative, \(f(x)\) will be \(|-x|+6\), which is the same as \(f(x)=|-x|+6\). If \(x\) is positive, \(f(x)\) will be \(|x|+6\), which is still the same as \(f(x)=|-x|+6\) because the absolute value function will negate the negative within its brackets.
03

Conclusion

Since the only difference between the two is the absolute value which negates any negatives, it can be concluded that the graphs of both \(f(x)=|x|+6\) and \(f(x)=|-x|+6\) are indeed identical.

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