Question

Determine the inverse of the function f (x) = 3(x − 4)^2 + 5.

Answer 1

Step-by-step explanation:

since g(f(x)) = x g (f (x) )= x, f - 1 (x) = 5x3 + 43 f - 1 (x) = 5 x 3 + 4 3

Answer 2

The inverse of the given function $f(x) = 3(x-4)^{2} +5$ is $f^{-1} (x) = \sqrt{\frac{x-5}{3} } +4$.

What is the inverse of a function?

An inverse function or an anti function is defined as a function, which can reverse into another function. In simple words, if any function “f” takes x to y then, the inverse of “f” will take y to x. If the function is denoted by ‘f’ or ‘F’, then the inverse function is denoted by $f^{-1}$ or $F^{-1}$.

According to the given equation.

We have a function.

$f(x) = 3(x-4)^{2} +5$

The above function can be written as

$y = 3(x-4)^{2} +5$

Write the above function for x.

$\implies y -5 = 3(x-4)^{2}$

$\implies \frac{y-5}{3} = (x-4)^{2}$

$\implies \sqrt{\frac{y-5}{3} } =x-4$

$\implies \sqrt{\frac{y-5}{3} } +4=x$

$0r\ x = \sqrt{\frac{y-5}{3}}+4$

Replace $x$ by $f^{-1}(x)$ and $y$ by $x$.

We get

$f^{-1} (x) = \sqrt{\frac{x-5}{3} } +4$

Hence, the inverse of the given function $f(x) = 3(x-4)^{2} +5$ is $f^{-1} (x) = \sqrt{\frac{x-5}{3} } +4$.

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