Questions: Question For the functions f(x)=2 x-2 and g(x)=6 x^2-3, find (g ∘ f)(x). Provide your answer below: (g ∘ f)(x)=

Transcript text: Question For the functions $f(x)=2 x-2$ and $g(x)=6 x^{2}-3$, find $(g \circ f)(x)$. Provide your answer below: \[ (g \circ f)(x)= \] $\square$ FEEDBACK MORE

Solution

Solution Steps

To find $(g \circ f)(x)$, we need to substitute $f(x)$ into $g(x)$. This means we will replace every $x$ in $g(x)$ with $f(x)$.

Step 1: Define the Functions

We have the functions defined as follows: \[ f(x) = 2x - 2 \] \[ g(x) = 6x^2 - 3 \]

Step 2: Compute $g(f(x))$

To find $(g \circ f)(x)$, we substitute $f(x)$ into $g(x)$: \[ g(f(x)) = g(2x - 2) \] Now, we replace $x$ in $g(x)$ with $2x - 2$: \[ g(2x - 2) = 6(2x - 2)^2 - 3 \]

Step 3: Simplify the Expression

First, we calculate $(2x - 2)^2$: \[ (2x - 2)^2 = 4x^2 - 8x + 4 \] Now substituting this back into $g$: \[ g(2x - 2) = 6(4x^2 - 8x + 4) - 3 \] Distributing the 6: \[ = 24x^2 - 48x + 24 - 3 \] Combining like terms: \[ = 24x^2 - 48x + 21 \]

Step 4: Evaluate at $x = 1$

Now, we evaluate $(g \circ f)(1)$: \[ (g \circ f)(1) = 24(1)^2 - 48(1) + 21 \] Calculating this gives: \[ = 24 - 48 + 21 = -3 \]