Questions: Determine whether (f) and (g) are inverse functions by evaluating ((f circ g)(x)) and ((g circ f)(x)). [f(x)=frac27 x-6 text and g(x)=frac7 x+422] What is ((f circ g)(x)) ? ((f circ g)(x)=) (square) (Use integers or fractions for any numbers in the expression.)

Solution Steps

To determine if \( f \) and \( g \) are inverse functions, we need to evaluate the compositions \( (f \circ g)(x) \) and \( (g \circ f)(x) \). If both compositions simplify to \( x \), then \( f \) and \( g \) are inverses of each other.

1. Compute \( (f \circ g)(x) = f(g(x)) \) by substituting \( g(x) \) into \( f(x) \).
2. Compute \( (g \circ f)(x) = g(f(x)) \) by substituting \( f(x) \) into \( g(x) \).
3. Check if both results simplify to \( x \).

To find \( (f \circ g)(x) \), substitute \( g(x) = \frac{7x + 42}{2} \) into \( f(x) = \frac{2}{7}x - 6 \):

\[ f(g(x)) = f\left(\frac{7x + 42}{2}\right) = \frac{2}{7} \left(\frac{7x + 42}{2}\right) - 6 \]

Simplifying the expression:

\[ = \frac{2}{7} \cdot \frac{7x + 42}{2} - 6 = \frac{7x + 42}{7} - 6 = x + 6 - 6 = x \]

Thus, \( (f \circ g)(x) = x \).

To find \( (g \circ f)(x) \), substitute \( f(x) = \frac{2}{7}x - 6 \) into \( g(x) = \frac{7x + 42}{2} \):

\[ g(f(x)) = g\left(\frac{2}{7}x - 6\right) = \frac{7\left(\frac{2}{7}x - 6\right) + 42}{2} \]

Simplifying the expression:

\[ = \frac{7 \cdot \frac{2}{7}x - 42 + 42}{2} = \frac{2x}{2} = x \]

Thus, \( (g \circ f)(x) = x \).

Final Answer