Which of the following statements are true about the graph of f left parenthesis x right parenthesis equals secant x? Select all that apply.

Question

(1 point)
Responses

left-parenthesis 0 comma 1 right-parenthesis is a point on the graph.
Image with alt text: left-parenthesis 0 comma 1 right-parenthesis is a point on the graph.

f left parenthesis x right parenthesis is defined for all x.
Image with alt text: f left parenthesis x right parenthesis is defined for all x .

There is a vertical asymptote at x equals Start Fraction pi over 2 End Fraction.
There is a vertical asymptote at Image with alt text: x equals Start Fraction pi over 2 End Fraction .

f left parenthesis x right parenthesis is undefined when sin x = 0.
Image with alt text: f left parenthesis x right parenthesis is undefined when sin x = 0.

All x-values are included in the domain.

Answer 1

To analyze the function \( f(x) = \sec(x) \), let's go through each statement provided:

1. \( (0, 1) \) is a point on the graph:
   We can evaluate \( f(0) \): \[ f(0) = \sec(0) = \frac{1}{\cos(0)} = \frac{1}{1} = 1. \] So, the point \( (0, 1) \) is indeed on the graph. (True)

2. \( f(x) \) is defined for all \( x \):
   The secant function is defined as \( \sec(x) = \frac{1}{\cos(x)} \). The function is undefined wherever \( \cos(x) = 0 \), which occurs at odd multiples of \( \frac{\pi}{2} \). Therefore, it is not defined for all \( x \). (False)

3. There is a vertical asymptote at \( x = \frac{\pi}{2} \):
   Since \( \cos\left(\frac{\pi}{2}\right) = 0 \), \( \sec\left(\frac{\pi}{2}\right) \) is undefined, and there is indeed a vertical asymptote at this point. (True)

4. \( f(x) \) is undefined when \( \sin x = 0 \):
   The secant function is related to cosine, not sine. The function is actually undefined when \( \cos x = 0 \), which is not necessarily when \( \sin x = 0 \). The values of \( x \) for which \( \sin x = 0 \) are \( n\pi \) (where \( n \) is an integer), and those points do not include vertical asymptotes (the function is defined there). (False)

5. All x-values are included in the domain:
   As discussed, since the secant function is undefined at odd multiples of \( \frac{\pi}{2} \), not all x-values are included in the domain. (False)

In summary, the true statements regarding the graph of \( f(x) = \sec(x) \) are:

• \( (0, 1) \) is a point on the graph.
• There is a vertical asymptote at \( x = \frac{\pi}{2} \).