QuestionAugust 27, 2025

11 Josie examines the graphs of f(x)=3^x-8 and g(x)=(1)/(x^2)-4 The number of solutions to f(x)=g(x) is (1) 1 (3) 3 (2) 2 (4) 0

11 Josie examines the graphs of f(x)=3^x-8 and g(x)=(1)/(x^2)-4 The number of solutions to f(x)=g(x) is (1) 1 (3) 3 (2) 2 (4) 0
11 Josie examines the graphs of f(x)=3^x-8 and g(x)=(1)/(x^2)-4 The number of solutions to f(x)=g(x) is (1) 1 (3) 3 (2) 2 (4) 0

Solution
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Answer

3 Explanation 1. Analyze the functions f(x) = 3^x - 8 is an exponential function shifted down by 8. g(x) = \frac{1}{x^2 - 4} is a rational function with vertical asymptotes at x = 2 and x = -2. 2. Determine behavior of f(x) As x \to \infty, f(x) \to \infty. As x \to -\infty, f(x) \to -8. 3. Determine behavior of g(x) As x \to 2^+ or x \to 2^-, g(x) \to \pm \infty. As x \to -2^+ or x \to -2^-, g(x) \to \pm \infty. As x \to \infty or x \to -\infty, g(x) \to 0. 4. Find intersections The graphs intersect where 3^x - 8 = \frac{1}{x^2 - 4}. Since f(x) is continuous and g(x) has discontinuities at x = 2 and x = -2, check intervals (-\infty, -2), (-2, 2), and (2, \infty). 5. Check intervals for solutions In (-\infty, -2), f(x) approaches -8 and g(x) approaches 0. They intersect once. In (-2, 2), f(x) increases from below -8 to above 0, while g(x) has a vertical asymptote. They intersect once. In (2, \infty), f(x) increases to infinity, g(x) decreases to 0. They intersect once.

Explanation

1. Analyze the functions<br /> $f(x) = 3^x - 8$ is an exponential function shifted down by 8. $g(x) = \frac{1}{x^2 - 4}$ is a rational function with vertical asymptotes at $x = 2$ and $x = -2$.<br /><br />2. Determine behavior of $f(x)$<br /> As $x \to \infty$, $f(x) \to \infty$. As $x \to -\infty$, $f(x) \to -8$.<br /><br />3. Determine behavior of $g(x)$<br /> As $x \to 2^+$ or $x \to 2^-$, $g(x) \to \pm \infty$. As $x \to -2^+$ or $x \to -2^-$, $g(x) \to \pm \infty$. As $x \to \infty$ or $x \to -\infty$, $g(x) \to 0$.<br /><br />4. Find intersections<br /> The graphs intersect where $3^x - 8 = \frac{1}{x^2 - 4}$. Since $f(x)$ is continuous and $g(x)$ has discontinuities at $x = 2$ and $x = -2$, check intervals $(-\infty, -2)$, $(-2, 2)$, and $(2, \infty)$.<br /><br />5. Check intervals for solutions<br /> In $(-\infty, -2)$, $f(x)$ approaches $-8$ and $g(x)$ approaches $0$. They intersect once.<br /> In $(-2, 2)$, $f(x)$ increases from below $-8$ to above $0$, while $g(x)$ has a vertical asymptote. They intersect once.<br /> In $(2, \infty)$, $f(x)$ increases to infinity, $g(x)$ decreases to $0$. They intersect once.
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