Question

For all values of x, f(x) = 7x² - 3, and for all values of y, g(y) = 2y + 9. What is g(f(x))?

• A. 7y² - 3
• B. 14x² + 3
• C. 14y² + 3
• D. 14x² - 3
• E. 2x + 9

Hint:

Don't be confused by the different variables \(x\) and \(y\). They are just placeholders for the inputs of their respective functions. When finding \(g(f(x))\), simply replace the input variable of the outer function \(g\) with the entire expression of the inner function \(f\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The notation \(g(f(x))\) represents the composition of two functions. It means we take the function \(f(x)\) and use its entire output as the input for the function \(g\). Note that the variable in \(g(y)\) is just a placeholder; \(g(y)=2y+9\) is the same rule as \(g(z)=2z+9\) or \(g(\text{input})=2(\text{input})+9\).
Step 2: Detailed Explanation:
We are given the functions:
\(f(x) = 7x^2 - 3\)
\(g(y) = 2y + 9\)
To find the composite function \(g(f(x))\), we substitute the expression for \(f(x)\) in place of the input variable \(y\) in the function \(g(y)\).
\[ g(f(x)) = g(7x^2 - 3) \] Now, we apply the rule for \(g\), which is to multiply the input by 2 and then add 9.
\[ g(f(x)) = 2(7x^2 - 3) + 9 \] Distribute the 2 across the terms in the parentheses.
\[ g(f(x)) = 2(7x^2) - 2(3) + 9 \] \[ g(f(x)) = 14x^2 - 6 + 9 \] Finally, combine the constant terms.
\[ g(f(x)) = 14x^2 + 3 \] Step 3: Final Answer:
The expression for the composite function \(g(f(x))\) is \(14x^2 + 3\).