1. **State the problem:** We have three functions: A) $f(x) = \frac{x^2 - 1}{x}$ B) $f(x) = x \sqrt{x^2 - 1}$ C) $f(x) = x^2 - 1$ We need to determine which function intersects the first axis (x-axis) at points $x=1$ and $x=-1$ but does **not** intersect the second axis (y-axis). 2. **Analyze each function for x-intercepts:** - The first axis (x-axis) is where $f(x) = 0$. For A: $$f(x) = \frac{x^2 - 1}{x} = \frac{(x-1)(x+1)}{x}$$ Setting $f(x)=0$ means numerator zero, denominator non-zero: $$x^2 - 1 = 0 \implies x=1 \text{ or } x=-1$$ These are valid x-intercepts since denominator $x \neq 0$ at these points. For B: $$f(x) = x \sqrt{x^2 - 1}$$ Set $f(x)=0$: $$x \sqrt{x^2 - 1} = 0$$ This is zero if either $x=0$ or $\sqrt{x^2 - 1} = 0$ which implies $x^2 - 1 = 0 \Rightarrow x = \pm 1$. At $x= \pm 1$, $f(x)=0$. For C: $$f(x) = x^2 - 1$$ Set $f(x)=0$: $$x^2 -1=0 \implies x=\pm 1$$ All three intersect x-axis at $x=\pm 1$ (except B also at 0). 3. **Analyze each function for y-intercepts:** The second axis (y-axis) is at $x=0$, so compute $f(0)$ where defined: For A: $$f(0) = \frac{0^2 - 1}{0} = \frac{-1}{0}$$ Not defined, so no y-intercept. For B: $$f(0) = 0 \times \sqrt{0^2 - 1} = 0 \times \sqrt{-1}$$ Not real-valued; no y-intercept. For C: $$f(0) = 0^2 - 1 = -1$$ So, crosses y-axis at $y=-1$. 4. **Conclusion:** - Only A and B do not have y-intercepts. - All three have x-intercepts at $x=\pm 1$ (B also at 0 but that is extra). Problem asks for function intersecting x-axis at $x=\pm 1$ and not intersecting y-axis. Therefore, functions A and B satisfy the condition. Since the question asks "Which of the following maps intersects the first axis $x=1$ and $x=-1$ but does not intersect the second axis?" The best fit is **A) $f(x) = \frac{x^2 - 1}{x}$**, because it has x-intercepts at $\pm 1$ and is undefined at zero, so no y-intercept. 5. **Summary:** - A) Intersects x-axis at $x=\pm 1$, no y-intercept. - B) Intersects x-axis at $x=\pm 1$ and also at 0; no real y-intercept. - C) Intersects both axes. **Final answer:** Function A.