If $g(x)$ is the inverse of $f(x)=x^3+2x+4$, evaluate $g(7)$

Question

Let $g(x)$ be the inverse of the function: $$f(x) = x^3+2x+4$$ Calculate $g(7)$.

I don't know where to begin, since you can't easily find the inverse by switching the $x$ and $y$ around.

Any and all help would be appreciated.

Answer 1

As KM101 commented, need to solve for $f(x)=7$. Since $g(y)$ is the inverse function, we know that whatever x value in $f(x)$ gives 7 is the output of $g(y)$.

$7=x^3+2x+4$

$0=x^3+2x-3$

$0=(x-1)*(x^2+x+3)$

$x=1$ There is no value of x that will make $0=(x^2+x+3)$ true so it will just be 1.

Answer 2

Hint. Obviously $f$ is an increasing function since all nonconstant terms have odd degree and positive coefficient, so it is invertible, and if we found $x$ so that $f(x)=7$ then it is the only one. Can you find such an $x$?

Answer 3

f(x)=7

x^3+2x+4=7

x^3+2x-3=0

(x-1)(x^2+x+3)=0

So,x=1is the answer. Thanks for asking.It was a very good question.