Questions: The graphs of y=f(x) (solid) and y=g(x) (dashed) are shown. Let h(x)=f(x)/g(x). h(x)=0 when x h(x) is positive in the interval (Enter your answer in interval notation) h(x) is negative in the interval (Enter your answer in interval notation)

Solution Steps

To find when \( h(x) = \frac{f(x)}{g(x)} = 0 \), we need to find when \( f(x) = 0 \) because a fraction is zero when its numerator is zero.

From the graph, \( f(x) \) (solid line) crosses the x-axis at \( x = -2 \).

\( h(x) = \frac{f(x)}{g(x)} \) is positive when both \( f(x) \) and \( g(x) \) have the same sign (both positive or both negative).

From the graph:

• \( f(x) \) is positive when \( x > -2 \).
• \( g(x) \) (dashed line) is positive when \( x < 1 \).

Thus, \( h(x) \) is positive in the interval \( (-2, 1) \).

\( h(x) = \frac{f(x)}{g(x)} \) is negative when \( f(x) \) and \( g(x) \) have opposite signs (one positive and one negative).

From the graph:

• \( f(x) \) is negative when \( x < -2 \).
• \( g(x) \) is negative when \( x > 1 \).

Thus, \( h(x) \) is negative in the intervals \( (-\infty, -2) \cup (1, \infty) \).

Final Answer