Question

If \(F(X) = X^2 - 2X\) and \(G(X) = 6X + 4\), which value of \(x\) makes \((FG)(X) = 0\)? - A) \(x = 2\) - B) \(x = -1\) - C) \(x = 0\) - D) \(x = -2\)

Answer 1

The value of x that makes
`\((FG)(x) = 0\)` is
`\(x = 2\)`. Thus the correct option is A.

Step-by-step explanation:

To find the value of x for which
`\((FG)(x) = 0\)`, we need to compute the product of the two given functions,
`\(F(x)\)` and G(x), and then solve for x when the product is equal to zero. The product of F(x) and G(x), denoted as
`\((FG)(x)\)`, is given by the expression
`\(F(x) \cdot G(x)\)`.

Given:

`\[ F(x) = X^2 - 2X \]`

`\[ G(x) = 6X + 4 \]`

The product \((FG)(x)\) is found by multiplying \(F(x)\) and \(G(x)\):

`\[ (FG)(x) = F(x) \cdot G(x) = (X^2 - 2X) \cdot (6X + 4) \]`

Setting
`\((FG)(x)\)` equal to zero:

`\[ (X^2 - 2X) \cdot (6X + 4) = 0 \]`

To find the values of \(x\), we can apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore:

`\[ X^2 - 2X = 0 \quad \text{or} \quad 6X + 4 = 0 \]`

Solving each equation individually:

`\[ X(X - 2) = 0 \quad \text{or} \quad 6X + 4 = 0 \]`

This leads to solutions X = 0 or X = 2 for the first equation, and
`\(X = -2/3\)` for the second equation. However, upon further inspection, the value X = 0 is invalid since it makes the second factor equal to 4, not zero. Thus, the correct value of \(x\) that satisfies
`\((FG)(x) = 0\)` is
`\(x = 2\)`, corresponding to option A.