Question

perform the following function operations given the functions $f(x) = x + 2$, $g(x) = 3x - 1$, and $h(x) = x^2 + 6x - 5$.
$(g + h)(x) = $
$(f \cdot g)(x) = $
$g(f(x)) = $
options: $3x + 5$, $3x^2 + 5x - 2$, $x^3 + 8x^2 + 7x - 5$, $x^2 + 7x - 3$, $4x^2 - 2$, $x^2 + 9x - 6$, $3x^2 + 16x - 16$, $3x^2$

Explanation:

Step1: Compute (g+h)(x)

Combine like terms of $g(x)$ and $h(x)$:
$$(g+h)(x) = (3x-1) + (x^2+6x-5) = x^2 + 9x - 6$$

Step2: Compute (f·g)(x)

Multiply $f(x)$ and $g(x)$:
$$(f \cdot g)(x) = (x+2)(3x-1) = 3x^2 -x +6x -2 = 3x^2 +5x -2$$

Step3: Compute g(f(x))

Substitute $f(x)$ into $g(x)$:
$$g(f(x)) = 3(x+2) -1 = 3x +6 -1 = 3x +5$$

Answer:

1. $(g+h)(x) = x^2 + 9x - 6$
2. $(f \cdot g)(x) = 3x^2 + 5x - 2$
3. $g(f(x)) = 3x + 5$