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Chapter 1: Problem 79

Sketch a graph of the function and determine whether it is even, odd, or neither. Verify your answer algebraically. $$f(x)=-|x-5|$$

Short Answer

Expert verified
After performing the necessary steps, it becomes clear that the graph of the function \(f(x) = -|x - 5|\) is a 'V' shape, opening downward, with the lowest point at (5, 0). In terms of the function鈥檚 parity, after substituting -x into the function and simplifying, the result is \(-|-x + 5|\), which is not the same as \(f(x) = -|x - 5|\), nor is it the same as \(-f(x)\). Hence, the function \(f(x) = -|x - 5|\) is neither even nor odd.

Step by step solution

01

Creating Table Of Values

A table of values is usually the first step to sketch a graph. Since the function is \(f(x) = -|x - 5|\), it can be divided into two different functions: \(f(x) = -(x - 5)\) for \(x \geq 5\) and \(f(x) = -(-x + 5)\) for \(x < 5\). For these cases, choose some values of x to find the corresponding values of y. For instance, for \(f(x) = -(x - 5)\), chosen values may be 5, 6, 7 and 8, and for \(f(x) = -(-x + 5)\), 4, 3, 2 and 1.
02

Sketching The Graph

On a graph paper, draw an x-axis and a y-axis. Each of these axes is a number line, and the point where they intersect represents zero on both lines. Then make a scatter plot using the table of values created in step 1. Each point on the graph represents an ordered pair (x, y). The vertical line, x = 5, splits the graph into two parts: one part is a line slanting upward to the left of x = 5, and another slanting downward to the right of x = 5. Connect the points to form a 'V'-shaped graph with the point (5, 0) at the bottom of the 'V'.
03

Determine Whether The Function Is Even, Odd Or Neither

The function is even if \(f(x) = f(-x)\) for every number x in the function's domain, and odd if \(f(-x) = -f(x)\) for every number x in the function's domain. In the case of our function, substitute -x into the function \(f(x) = -|x - 5|\) and simplify. Compare the simplified function to the original function \(f(x) = -|x - 5|\). If they are the same, then the function is even; if they differ only by a sign, then the function is odd. If they differ in any other way, then the function is neither even nor odd.

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