Question

a composition of functions is defined as (f(g(x)) = x - 3).
which of the following could be expressions for (f) and (g)?

• (f(x) = sqrt{x} - 5) and (g(x) = (x + 2)^2)
• (f(x) = sqrt{x} - 5) and (g(x) = x^2 + 2)
• (f(x) = sqrt{x - 5}) and (g(x) = (x + 2)^2)
• (f(x) = sqrt{x - 5}) and (g(x) = x^2 + 2)

Explanation:

⚡ Using what you learned: Evaluating Composite Functions

Step 1: Analyze Option 1 and Option 3

Evaluate \( f(g(x)) \) for \( f(x) = \sqrt{x} - 5 \) and \( g(x) = (x+2)^2 \):

$$ f(g(x)) = \sqrt{(x+2)^2} - 5 $$

Assuming \( x+2 \ge 0 \):

$$ f(g(x)) = (x+2) - 5 = x - 3 $$

Step 2: Analyze Option 2 and Option 4

Evaluate \( f(g(x)) \) for \( f(x) = \sqrt{x} - 5 \) and \( g(x) = x^2 + 2 \):

$$ f(g(x)) = \sqrt{x^2+2} - 5 eq x - 3 $$

Step 3: Identify the correct option

Looking closely at the image options:

• Option 1: \( f(x) = \sqrt{x} - 5 \) and \( g(x) = (x+2)^2 \)
• Option 2: \( f(x) = \sqrt{x} - 5 \) and \( g(x) = x^2 + 2 \)
• Option 3: \( f(x) = \sqrt{x-5} \) and \( g(x) = (x+2)^2 \)
• Option 4: \( f(x) = \sqrt{x-5} \) and \( g(x) = x^2 + 2 \)

Let's test Option 3:

$$ f(g(x)) = \sqrt{(x+2)^2 - 5} eq x - 3 $$

Let's test Option 4:

$$ f(g(x)) = \sqrt{(x^2+2) - 5} = \sqrt{x^2 - 3} eq x - 3 $$

Thus, only Option 1 yields \( f(g(x)) = x - 3 \).

Answer:

\( f(x) = \sqrt{x} - 5 \) and \( g(x) = (x + 2)^2 \) (the first option)