consider the function 8e^(3x + 2) - 2. what input value corresponds to f(x) = 6? round your answer to two…

Explanation:

Step1: Set up the equation

Set $8e^{(3x + 2)}-2=6$.

Step2: Isolate the exponential term

First, add 2 to both sides of the equation: $8e^{(3x + 2)}=6 + 2=8$. Then divide both sides by 8, we get $e^{(3x + 2)} = 1$.

Step3: Take the natural - logarithm of both sides

Since $y = e^x$ and $y=\ln x$ are inverse functions, $\ln(e^{(3x + 2)})=\ln(1)$. Using the property $\ln(e^a)=a$, we have $3x + 2=\ln(1)$.

Step4: Solve for x

We know that $\ln(1)=0$, so the equation becomes $3x+2 = 0$. Subtract 2 from both sides: $3x=-2$. Then divide by 3: $x=-\frac{2}{3}\approx - 0.67$.

Answer:

$-0.67$