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Chapter 1: Problem 81

Sketch a graph of the function and determine whether it is even, odd, or neither. Verify your answer algebraically. $$f(x)=\sqrt{1-x}$$

Short Answer

Expert verified
The function \(f(x) = \sqrt{1-x}\) is neither even nor odd.

Step by step solution

01

Graphing the function

Firstly plot the function to get a visual representation of it. The graph of \(f(x) = \sqrt{1-x}\) is recognized as half of an upper unit circle shifted to the right by one unit.
02

Determining even, odd or neither

The graph achieved from Step 1 does not show symmetry about the y-axis or origin; therefore, the function is neither even nor odd visually.
03

Verifying algebraically

To verify this algebraically, substitute \(-x\) for \(x\) in the equation to see if we get \(f(x)\) or \(-f(x)\). On performing the substitution \(f(-x) = \sqrt{1-(-x)} = \sqrt{1+x}\). This is not equal to \(f(x)\) nor \(-f(x)\). So, the function is neither even nor odd algebraically as well.

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