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Two functions, f and g, are defined over all real numbers by f(x)=2x-1 (September 2023 - G01)
Two functions, f and g, are defined over all real numbers by f(x)=2x-1 and g(x)=3x^2-7.
What is g(f(5x)) ?
A. 150x^2-15
B. 300x^2-60x-4
C. 60x^3-60x^2-20x
D. 750x^3-75x^2-70x+7
E. 30x^4-15x^3-70x^2+35x
Solution
Correct answer: B. 300x^2-60x-4.
The question asks for the value of g(f(5x)) which involves applying two functions one after another. Let's break it down step by step:
Inner Function (f(5x)): Here, we evaluate the inner function f(x) = 2x-1, but with x replaced by 5x. So, f(5x) = 2*(5x) - 1 = 10x - 1.
Outer Function (g(f(5x))): Now, we take the result of the inner function (10x-1) and substitute it into the outer function g(x) = 3x^2-7. So, g(f(5x)) = g(10x-1)
Evaluate g(10x-1): Finally, we evaluate the outer function g(x) = 3x^2-7 with x replaced by (10x-1). Therefore:
g(10x-1) = 3 * [(10x-1)]^2 - 7
Expanding the square in the brackets:
g(10x-1) = 3 * (100x^2 - 20x + 1) - 7
Multiplying terms inside the parenthesis:
g(10x-1) = 300x^2 - 60x + 3 - 7
Combining terms:
g(10x-1) = 300x^2 - 60x - 4
Exam Edition: September 2023 - G01
Exam Year: 2023
Related topics: Functions and Function Composition, Algebraic Manipulations