Questions: Find the composite function for the given functions. g ◦ f for f(x) = (x-7)/3 and g(x) = 3x+7

Solution Steps

To find the composite function \( g \circ f \), we need to substitute the function \( f(x) \) into the function \( g(x) \). This means we will replace every instance of \( x \) in \( g(x) \) with \( f(x) \).

We are given two functions: \[ f(x) = \frac{x - 7}{3} \] \[ g(x) = 3x + 7 \]

To find the composite function \( g \circ f \), we substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g\left(\frac{x - 7}{3}\right) \]

Substitute \( \frac{x - 7}{3} \) into \( g(x) \): \[ g\left(\frac{x - 7}{3}\right) = 3 \left(\frac{x - 7}{3}\right) + 7 \]

Simplify the expression: \[ g\left(\frac{x - 7}{3}\right) = x - 7 + 7 \] \[ g\left(\frac{x - 7}{3}\right) = x \]

To verify, we can use \( x = 10 \): \[ f(10) = \frac{10 - 7}{3} = 1 \] \[ g(f(10)) = g(1) = 3 \cdot 1 + 7 = 10 \]

Final Answer