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Select the correct answer.

Consider this function and its graph:
[tex] f(x) = \tan \left(3x + \frac{\pi}{4}\right) - 1 [/tex]

Which statement best describes the function [tex] f(x) [/tex]?

A. The function is odd.
B. There is not enough information to determine whether the function is even or odd.
C. The function is neither even nor odd.
D. The function is even.


Sagot :

To determine whether the function [tex]\(f(x) = \tan \left(3x + \frac{\pi}{4}\right) - 1\)[/tex] is even, odd, or neither, we must check how it behaves under transformations involving the negative of its argument. Specifically, we will analyze [tex]\(f(-x)\)[/tex]:

1. Define the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = \tan(3x + \frac{\pi}{4}) - 1 \][/tex]

2. Find [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = \tan(3(-x) + \frac{\pi}{4}) - 1 = \tan(-3x + \frac{\pi}{4}) - 1 \][/tex]

3. Simplify [tex]\( \tan(-3x + \frac{\pi}{4}) \)[/tex]:
Using the property of the tangent function:
[tex]\[ \tan(-\theta) = -\tan(\theta) \][/tex]
Therefore,
[tex]\[ \tan(-3x + \frac{\pi}{4}) = -\tan(3x - \frac{\pi}{4}) \][/tex]

4. Substitute back to get [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = -\tan(3x - \frac{\pi}{4}) - 1 \][/tex]

5. Compare [tex]\( f(-x) \)[/tex] and [tex]\( f(x) \)[/tex]:
We now compare [tex]\( f(-x) \)[/tex] to both [tex]\( f(x) \)[/tex] and [tex]\(-f(x)\)[/tex] to determine if the function is even, odd, or neither.

- A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex].
[tex]\[ f(-x) = -\tan(3x - \frac{\pi}{4}) - 1 \quad \text{and} \quad f(x) = \tan(3x + \frac{\pi}{4}) - 1 \][/tex]
Clearly, [tex]\[f(-x) \neq f(x)\][/tex]

- A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex].
[tex]\[ -f(x) = - (\tan(3x + \frac{\pi}{4}) - 1) = -\tan(3x + \frac{\pi}{4}) + 1 \][/tex]
and,
[tex]\[ f(-x) = -\tan(3x - \frac{\pi}{4}) - 1 \][/tex]
Clearly, [tex]\[f(-x) \neq -f(x)\][/tex]

Since [tex]\( f(-x) \)[/tex] is neither [tex]\( f(x) \)[/tex] nor [tex]\(-f(x)\)[/tex], we can conclude that the function [tex]\( f(x) \)[/tex] is neither even nor odd.

Conclusion:
The correct statement is:
C. The function is neither even nor odd.
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