The function 𝑓 is given by 𝑓(𝑥)=𝑒2𝑥, and the function 𝑔 is given by 𝑔(𝑥)=ln(3𝑥). For 𝑥>0, which of the following is an expression for 𝑓(𝑔(𝑥)) ? Responses9𝑥29 x squared2𝑥+ln32 x plus ln 3(𝑒2𝑥)·ln(3𝑥)open parentheses e to the power of 2 x end exponent close parentheses times ln open parentheses 3 x close parentheses
Question
The function 𝑓 is given by 𝑓(𝑥)=𝑒2𝑥, and the function 𝑔 is given by 𝑔(𝑥)=ln(3𝑥). For 𝑥>0, which of the following is an expression for 𝑓(𝑔(𝑥)) ? Responses9𝑥29 x squared2𝑥+ln32 x plus ln 3(𝑒2𝑥)·ln(3𝑥)open parentheses e to the power of 2 x end exponent close parentheses times ln open parentheses 3 x close parentheses
Solution
To find the expression for 𝑓(𝑔(𝑥)), we need to substitute 𝑔(𝑥) into 𝑓(𝑥).
Given 𝑓(𝑥) = 𝑒^(2𝑥) and 𝑔(𝑥) = ln(3𝑥), we substitute 𝑔(𝑥) into 𝑓(𝑥) to get:
𝑓(𝑔(𝑥)) = 𝑓(ln(3𝑥)) = 𝑒^(2ln(3𝑥))
Using the property of logarithms that allows us to move the coefficient of the logarithm up as the power of the argument, we get:
𝑒^(ln((3𝑥)^2)) = (3𝑥)^2
So, 𝑓(𝑔(𝑥)) = (3𝑥)^2, which is equivalent to 9𝑥^2. Therefore, the correct answer is 9𝑥^2.
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