Sets & FunctionsMedium

Inverse function — exponential and logarithm

Given f(x) = e^{3x−2}, find the inverse function f⁻¹(x) and its domain.

Given data

f(x) = e^{3x−2}

Steps

0 / 4

Write y = f(x) — what operation should be applied to isolate x?
Write your working — the numerical result must appear in the text (logica)
Apply ln to both sides: ln(y) = ?
Isolate x: x = ?
Swapping x and y, what is f⁻¹(x)? (verification question)

Full worked solution

Write y = f(x) — what operation should be applied to isolate x?
$y = e^{3x-2}$
Set $y = e^{3x-2}$ . To isolate $x$ , we apply the natural logarithm $\ln$ to both sides, which is the inverse of the exponential function.
Apply ln to both sides: ln(y) = ?
$\ln y = 3x - 2$
$\ln y = \ln(e^{3x-2}) = 3x - 2$ . The logarithm cancels the exponential, leaving the exponent.
Isolate x: x = ?
$x = \dfrac{\ln y + 2}{3}$
$x = (\ln y + 2)/3$ . Solving $\ln y = 3x - 2$ for $x$ gives $x = (\ln y + 2)/3$ .
Swapping x and y, what is f⁻¹(x)? (verification question)
$f^{-1}(x) = \dfrac{\ln x + 2}{3}$
$f^{-1}(x) = (\ln x + 2)/3$ . The domain of $f^{-1}$ is $(0, +\infty)$ because $\ln x$ requires $x > 0$ , which matches the range of $f$ .

Result:f⁻¹(x) = (ln x + 2)/3, domain (0, +∞).

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