Questions: For the given functions f and g , find the indicated composition. f(x)=(x-6)/2, g(x)=2x+6 ;(g∘f)(x) A. 2x+6 B. x-3 C. x+12 D. x

Solution Steps

To find the composition \((g \circ f)(x)\), 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 \(f(x)\). After substitution, simplify the expression to find the resulting function.

We have the functions defined as follows: \[ f(x) = \frac{x - 6}{2} \] \[ g(x) = 2x + 6 \]

To find \((g \circ f)(x)\), we substitute \(f(x)\) into \(g(x)\): \[ g(f(x)) = g\left(\frac{x - 6}{2}\right) \] Substituting \(f(x)\) into \(g(x)\): \[ g\left(\frac{x - 6}{2}\right) = 2\left(\frac{x - 6}{2}\right) + 6 \]

Now, we simplify the expression: \[ = (x - 6) + 6 = x \]

Final Answer