Certainly! Let's analyze the function step-by-step.
Given function:
[tex]\[ f(x) = |2x - 5| - 1 \][/tex]
(a) Test for symmetry with respect to both axes and the origin:
1. Symmetry with respect to the y-axis:
A function [tex]\( f(x) \)[/tex] is symmetric with respect to the y-axis if [tex]\( f(-x) = f(x) \)[/tex].
Let’s check [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = |2(-x) - 5| - 1 = |-2x - 5| - 1 \][/tex]
For [tex]\( f(x) = |2x - 5| - 1 \)[/tex]:
We need to compare [tex]\( |-2x - 5| - 1 \)[/tex] with [tex]\( |2x - 5| - 1 \)[/tex].
Clearly, [tex]\(|2x - 5|\)[/tex] and [tex]\(|-2x - 5|\)[/tex] are not necessarily equal for all x, so:
[tex]\[ f(-x) \neq f(x) \][/tex]
Thus, the function is not symmetric with respect to the y-axis.
2. Symmetry with respect to the origin:
A function [tex]\( f(x) \)[/tex] is symmetric with respect to the origin if [tex]\( f(-x) = -f(x) \)[/tex].
We already have [tex]\( f(-x) = |-2x - 5| - 1 \)[/tex].
Let’s compare [tex]\( f(-x) = |-2x - 5| - 1 \)[/tex] with [tex]\(-f(x) = -(|2x - 5| - 1)\)[/tex]:
[tex]\[ -f(x) = -(|2x - 5| - 1) = -|2x - 5| + 1 \][/tex]
Clearly, [tex]\(|-2x - 5|\)[/tex] and [tex]\(|2x - 5|\)[/tex] are not equal in magnitude for all x, and also [tex]\(|-2x - 5| - 1\)[/tex] does not equal [tex]\(-|2x - 5| + 1\)[/tex]. Therefore:
[tex]\[ f(-x) \neq -f(x) \][/tex]
Thus, the function is not symmetric with respect to the origin.
(b) Determine if the equation is even, odd, or neither:
Based on our symmetry tests:
- The function does not satisfy the condition for being even (symmetric with respect to y-axis).
- The function does not satisfy the condition for being odd (symmetric with respect to the origin).
Since the function is neither symmetric with respect to the y-axis nor the origin:
[tex]\[ f(x) \text{ is neither even nor odd} \][/tex]
Thus, the detailed analysis leads us to the conclusion:
1. [tex]\( f(x) \)[/tex] is not symmetric with respect to the y-axis.
2. [tex]\( f(x) \)[/tex] is not symmetric with respect to the origin.
3. The function is neither even nor odd.
Therefore, the final results are:
- Symmetry with respect to y-axis: False
- Symmetry with respect to origin: False
- Even: False
- Odd: False
- Neither: True
