You are given the following two functions:

$f(x) = 2x + 3$ $g(x) = x^2 - x + 2$ Additionally, you are provided that $g(a) = 4$, and you need to find $(g \circ f)(x)$, which is the composition of $g$ and $f$, denoted by $gof$.

To solve $(g \circ f)(x)$, we first substitute $f(x)$ into $g(x)$:

$(g \circ f)(x) = g(f(x))$

1. First, find $f(x)$: $f(x) = 2x + 3$

2. Now, substitute $f(x)$ into $g(x)$: $g(f(x)) = g(2x + 3) = (2x + 3)^2 - (2x + 3) + 2$

3. Simplify each part:

• Expand $(2x + 3)^2$: $(2x + 3)^2 = 4x^2 + 12x + 9$

• Now substitute it into $g(f(x))$: $g(f(x)) = (4x^2 + 12x + 9) - (2x + 3) + 2$ $= 4x^2 + 12x + 9 - 2x - 3 + 2$ $= 4x^2 + 10x + 8$

Thus, $(g \circ f)(x) = 4x^2 + 10x + 8$.

Now, regarding $g(a) = 4$:

This means: $g(a) = a^2 - a + 2 = 4$

Solving for $a$:

$a^2 - a + 2 = 4$ $a^2 - a - 2 = 0$

Factor the quadratic equation: $(a - 2)(a + 1) = 0$

So, $a = 2$ or $a = -1$.

Would you like more details on the composition or solving the equation?

Related Questions:

1. What is the meaning of function composition in mathematics?
2. How do you expand the square of a binomial?
3. Can a function have more than one value of $x$ that satisfies $g(x) = 4$?
4. How do you factor a quadratic equation?
5. What is the domain of the composite function $(g \circ f)(x)$?

Tip:

Always check the domain of the inner function when performing function composition to ensure the result is valid for all $x$.