Question

If the functions \( f:\mathbb{R} \to \mathbb{R} \) and \( g:\mathbb{R} \to \mathbb{R} \) are defined as \( f(x)=\cos x \) and \( g(x)=3x^2 \) respectively then find gof.

Hint:

Be careful with the order of composition. \( g \circ f \) is not the same as \( f \circ g \). For example, \( (f \circ g)(x) = f(g(x)) = f(3x^2) = \cos(3x^2) \), which is a different function.

Answer 1

Step 1: Understanding the Concept:
The notation 'gof' represents the composition of functions, which is read as "g of f". It means we first apply the function f to x, and then apply the function g to the result of f(x).
Step 2: Key Formula or Approach:
The composition of functions \( g \circ f \) is defined as: \[ (g \circ f)(x) = g(f(x)) \] Step 3: Detailed Explanation:
We are given the functions: \[ f(x) = \cos x \] \[ g(x) = 3x^2 \] To find \( (g \circ f)(x) \), we substitute \( f(x) \) into \( g(x) \): \[ (g \circ f)(x) = g(f(x)) = g(\cos x) \] Now, we apply the function \( g \) to the input \( \cos x \). The rule for \( g \) is to take the input, square it, and multiply by 3. \[ g(\cos x) = 3(\cos x)^2 = 3\cos^2 x \] Step 4: Final Answer:
The composite function is \( (g \circ f)(x) = 3\cos^2 x \).