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 $x=\frac{\pi }{2}$.