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Chapter 1: Problem 86

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$ f(t)=t^{2}+2 t-3 $$

Short Answer

Expert verified
The function \( f(t) = t^{2} + 2t - 3 \) is neither even nor odd. It does not have symmetry with respect to the y-axis or the origin.

Step by step solution

01

Identify Function

We are given the function \( f(t) = t^{2} + 2t - 3 \).
02

Test for Even Symmetry

To see if the function is even, we replace 't' with '-t' in the function: \( f(-t) = (-t)^{2} + 2(-t) - 3 = t^{2} - 2t - 3 \). If \( f(t) = f(-t) \), then the function is even. In this case, \( f(t) \neq f(-t) \), so the function is not even.
03

Test for Odd Symmetry

To see if the function is odd, we find -f(t) and compare it to f(-t). -f(t) is -t^{2} - 2t + 3. Comparing this to f(-t), which we found in the last step to be t^{2} - 2t - 3, we can see that \( -f(t) \neq f(-t) \) also. Thus, the function is also not odd.
04

Describe Symmetry

Since the function is neither even nor odd, it does not have symmetry with respect to the y-axis or the origin.

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