Question

Let $f(x) = x^2 - 10x$ and $g(x) = e^x + 5$ for $x \in \mathbb{R}$. Then, for all $x$, $g(2x) - (f \circ g)(x) = $}

• A. 30
• B. 20
• C. -30
• D. -20
• E. 15

Hint:

Notice that \( f(x) = (x-5)^2 - 25 \). Substituting \( g(x) = e^x + 5 \) makes \( f(g(x)) = ((e^x+5)-5)^2 - 25 = e^{2x} - 25 \). This "completing the square" method can save significant time.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires calculating the value of a composite function and manipulating algebraic expressions involving exponents.
Key Formula or Approach:
1. Evaluate \( g(2x) \) by substituting $2x$ for $x$ in $g(x)$.
2. Evaluate \( (f \circ g)(x) \) by substituting the expression for \( g(x) \) into \( f(x) \).
3. Subtract the two resulting expressions.

Step 2: Detailed Explanation:
Given \( f(x) = x^2 - 10x \) and \( g(x) = e^x + 5 \).
Calculate \( g(2x) \):
\[ g(2x) = e^{2x} + 5 \]
Calculate \( (f \circ g)(x) \):
\[ f(g(x)) = f(e^x + 5) \]
Substitute \( (e^x + 5) \) into \( f(x) = x^2 - 10x \):
\[ (e^x + 5)^2 - 10(e^x + 5) \]
Expand the square \( (a+b)^2 = a^2 + 2ab + b^2 \):
\[ (e^x)^2 + 2(5)(e^x) + 5^2 - 10e^x - 50 \]
\[ e^{2x} + 10e^x + 25 - 10e^x - 50 \]
Simplify by cancelling \( 10e^x \):
\[ e^{2x} + 25 - 50 = e^{2x} - 25 \]
Now, calculate the final expression:
\[ g(2x) - (f \circ g)(x) = (e^{2x} + 5) - (e^{2x} - 25) \]
\[ e^{2x} + 5 - e^{2x} + 25 = 30 \]

Step 3: Final Answer:
The result is 30.