Find \( (f \circ g)(x) \) and \( (g \circ f)(x) \) and graph each of these functions. \( f(x)=-5 x \) \( g(x)=\cos x \) \( (f \circ g)(x)=-5 \cos x \) \( ( \) Simplify your answer.) \( (g \circ f)(x)=\cos (5 x) \) (Simplify your answer.) Use the graphing tool to graph the functions of \( (f \circ g)(x) \) and \( (g \circ f)(x) \) :

To find the compositions of the functions \( f \) and \( g \), we have: 1. **Finding \( (f \circ g)(x) \)**: \[ (f \circ g)(x) = f(g(x)) = f(\cos x) \] Since \( f(x) = -5x \), we substitute \( \cos x \) into \( f \): \[ f(\cos x) = -5(\cos x) = -5 \cos x \] Therefore, \[ (f \circ g)(x) = -5 \cos x \] 2. **Finding \( (g \circ f)(x) \)**: \[ (g \circ f)(x) = g(f(x)) = g(-5x) \] Since \( g(x) = \cos x \), we substitute \( -5x \) into \( g \): \[ g(-5x) = \cos(-5x) \] Using the property of cosine that \( \cos(-\theta) = \cos(\theta) \), we have: \[ \cos(-5x) = \cos(5x) \] Therefore, \[ (g \circ f)(x) = \cos(5x) \] ### Summary of Results: - \( (f \circ g)(x) = -5 \cos x \) - \( (g \circ f)(x) = \cos(5x) \) ### Graphing the Functions: To graph these functions, you can use graphing software or a graphing calculator. Here’s a brief description of what to expect from the graphs: 1. **Graph of \( (f \circ g)(x) = -5 \cos x \)**: - This function is a cosine wave with an amplitude of 5 (since it is multiplied by -5, it will be inverted). - The period remains \( 2\pi \). - The graph oscillates between 5 and -5. 2. **Graph of \( (g \circ f)(x) = \cos(5x) \)**: - This function is a cosine wave with a period of \( \frac{2\pi}{5} \) (since the argument is multiplied by 5). - The amplitude is 1. - The graph oscillates between 1 and -1, with more oscillations in the same interval compared to the standard cosine function. You can plot these functions over a suitable range, such as \( x \) from \( -2\pi \) to \( 2\pi \), to see their behavior.