Question

If f(x)=3x+7 and g(x)=x²-12, what is f(g(x))?

• A. 3x³-29
• B. 9x²-29
• C. 3x²+29
• D. 3x²-29
• E. 9x³+29

Hint:

Be careful with the order of operations. In \(f(g(x))\), \(g(x)\) is the input for \(f\). If the question asked for \(g(f(x))\), you would substitute \(f(x)\) into \(g(x)\), which would yield a different result: \((3x+7)^2 - 12\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The notation \(f(g(x))\) represents a composite function. To find the expression for \(f(g(x))\), we substitute the entire expression for the inner function, \(g(x)\), into every instance of \(x\) in the outer function, \(f(x)\).
Step 2: Detailed Explanation:
We are given the functions:
\(f(x) = 3x + 7\)
\(g(x) = x^2 - 12\)
To find \(f(g(x))\), we take the expression for \(g(x)\) and plug it into \(f(x)\).
\[ f(g(x)) = f(x^2 - 12) \] Now, in the function \(f(x) = 3x + 7\), we replace \(x\) with \((x^2 - 12)\).
\[ f(g(x)) = 3(x^2 - 12) + 7 \] Next, we simplify the expression by distributing the 3.
\[ f(g(x)) = 3x^2 - 3(12) + 7 \] \[ f(g(x)) = 3x^2 - 36 + 7 \] Finally, combine the constant terms.
\[ f(g(x)) = 3x^2 - 29 \] Step 3: Final Answer:
The composite function \(f(g(x))\) is \(3x^2 - 29\).