Question:
If \( f(x) = x^2 - 10x \), \( g(x) = e^x + 5 \), find \( g(2x) - f(g(x)) \).
If \( f(x) = x^2 - 10x \), \( g(x) = e^x + 5 \), find \( g(2x) - f(g(x)) \).
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When dealing with composite functions, always substitute one function into the other carefully and simplify the expression step by step.
Updated On: Apr 18, 2026
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Solution and Explanation
Step 1: Find \( g(2x) \).
We are given \( g(x) = e^x + 5 \). To find \( g(2x) \), substitute \( 2x \) into \( g(x) \): \[ g(2x) = e^{2x} + 5 \]
Step 2: Find \( f(g(x)) \).
We are given \( f(x) = x^2 - 10x \). To find \( f(g(x)) \), substitute \( g(x) = e^x + 5 \) into \( f(x) \): \[ f(g(x)) = (e^x + 5)^2 - 10(e^x + 5) \] Expanding the terms: \[ f(g(x)) = (e^{2x} + 10e^x + 25) - (10e^x + 50) \] Simplifying: \[ f(g(x)) = e^{2x} + 25 - 50 = e^{2x} - 25 \]
Step 3: Find \( g(2x) - f(g(x)) \).
Now, we subtract \( f(g(x)) \) from \( g(2x) \): \[ g(2x) - f(g(x)) = (e^{2x} + 5) - (e^{2x} - 25) \] Simplifying: \[ g(2x) - f(g(x)) = e^{2x} + 5 - e^{2x} + 25 = 30 \]
We are given \( g(x) = e^x + 5 \). To find \( g(2x) \), substitute \( 2x \) into \( g(x) \): \[ g(2x) = e^{2x} + 5 \]
Step 2: Find \( f(g(x)) \).
We are given \( f(x) = x^2 - 10x \). To find \( f(g(x)) \), substitute \( g(x) = e^x + 5 \) into \( f(x) \): \[ f(g(x)) = (e^x + 5)^2 - 10(e^x + 5) \] Expanding the terms: \[ f(g(x)) = (e^{2x} + 10e^x + 25) - (10e^x + 50) \] Simplifying: \[ f(g(x)) = e^{2x} + 25 - 50 = e^{2x} - 25 \]
Step 3: Find \( g(2x) - f(g(x)) \).
Now, we subtract \( f(g(x)) \) from \( g(2x) \): \[ g(2x) - f(g(x)) = (e^{2x} + 5) - (e^{2x} - 25) \] Simplifying: \[ g(2x) - f(g(x)) = e^{2x} + 5 - e^{2x} + 25 = 30 \]
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