The function $h(x)=\frac{4}{3} x-1$ represents the composite $h(x)=f(g(x))$. If $f(x)=2 x-1$, what is $g(x)$ ?
A. $g(x)=\frac{2}{3} x$
B. $g(x)=\frac{2}{3} x-1$
C. $g(x)=2 \frac{1}{3} x$
D. $g(x)=2 \frac{1}{3} x-1$
Answer (2)
We have f ( g ( x )) = h ( x ) where f ( x ) = 2 x − 1 and h ( x ) = 3 4 x − 1 .
Substitute g ( x ) into f ( x ) to get 2 g ( x ) − 1 = 3 4 x − 1 .
Add 1 to both sides: 2 g ( x ) = 3 4 x .
Divide by 2: g ( x ) = 3 2 x .
The final answer is g ( x ) = 3 2 x .
Explanation
Understanding the Problem We are given the composite function h ( x ) = f ( g ( x )) = 3 4 x − 1 and the function f ( x ) = 2 x − 1 . Our goal is to find the function g ( x ) .
Setting up the Equation Since h ( x ) = f ( g ( x )) , we can write f ( g ( x )) = 2 g ( x ) − 1 . We also know that h ( x ) = 3 4 x − 1 . Therefore, we can set up the equation 2 g ( x ) − 1 = 3 4 x − 1 .
Isolating the Term with g(x) To solve for g ( x ) , we first add 1 to both sides of the equation: 2 g ( x ) − 1 + 1 = 3 4 x − 1 + 1
2 g ( x ) = 3 4 x
Solving for g(x) Next, we divide both sides of the equation by 2: 2 2 g ( x ) = 2 3 4 x
g ( x ) = 3 4 x ⋅ 2 1
Simplifying the Expression Finally, we simplify the expression for g ( x ) : g ( x ) = 6 4 x = 3 2 x
Final Answer Therefore, g ( x ) = 3 2 x .
Examples
Imagine you're baking a cake, and f ( x ) represents the baking process that doubles the amount of ingredients x and then reduces it by one unit (perhaps due to some loss during baking). If you want the final cake h ( x ) to be 3 4 x − 1 , you need to figure out the initial amount of ingredients g ( x ) that you should prepare. This problem helps you determine that initial amount, ensuring the final cake has the desired quantity. Understanding function composition helps in planning processes where multiple steps are involved to achieve a specific outcome.
The function g ( x ) is found to be g ( x ) = 3 2 x , which corresponds to option A. This was determined by using the relationship between h ( x ) and the function f ( x ) . Algebraic manipulation led to isolating g ( x ) .
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