Question

What is (f+g)(x)?

A. |3x-5| +7x - 2x + 9
B. |3x-5| +5x + 9
C. |3x-5| +9x + 4
D. |3x-5| +5x - 2

Answer 1

C. |3x-5| +9x + 4

Step-by-step explanation:

To find (f+g)(x), we need to add the two functions, f(x) and g(x). Let's express the functions f(x) and g(x) first:

`\[f(x) = |3x-5|\]`

`\[g(x) = 5x + 4\]`

Now, to find (f+g)(x), we add these functions:

`\[(f+g)(x) = f(x) + g(x) = |3x-5| + (5x + 4)\]`

Now, let's simplify the expression. The absolute value function |3x-5| has two cases based on the inside expression:

1. When
`\(3x-5 \geq 0\): \(|3x-5| = 3x-5\)`

2. When
`\(3x-5 < 0\): \(|3x-5| = -(3x-5)\)`

Combine these cases with the other terms:

`\[(f+g)(x) = (3x-5 + 5x + 4) \text{ if } (3x-5 \geq 0)\]`

`\[(f+g)(x) = (-(3x-5) + 5x + 4) \text{ if } (3x-5 < 0)\]`

Now simplify each case:

`\[(f+g)(x) = 8x - 1 \text{ if } (3x-5 \geq 0)\]`

`\[(f+g)(x) = 14x + 9 \text{ if } (3x-5 < 0)\]`

This creates a piecewise function. However, looking at the answer choices, it seems there might be an error in the options provided. None of the given options match the correct expression for (f+g)(x). It's possible there might be a mistake in the formulation of the answer choices. Please double-check the provided options.