How to sketch the graph of an inverse function?

Question

Given the function $f(x)=x^2$

i) Write $f^{-1}(x)$ in the form of $y$

ii) Sketch the graph $f^{-1}(x)$ and $f^{-1}(x+1)$ on the same set of axes

iii) Use your graphs to solve for $x$ if $\log_2(x+2) < 1$

Now looking at the solution, the graph shows that the Inverse function going through the point $(1,0)$ so what i don't understand is that how did they get to the mentioned point if no value of $x$ or $y$ is given (so I could plug $y$ OR $x$ into the equation and and find $x$ & $y$ values in order to sketch the graph?

That's weird. The inverse function of $f(x) = x^2$ doesn't pass through $(1,0)$ because $0^2 = 0 \neq 1$ ... — Matti P., Apr 10 '18 at 10:07
Also the question is unclear. Usually, in order to sketch the graph of a function, you can choose the values of the independent variable (usually $x$) freely. It's best to choose "easy" values, for example $0$ and $1$, etc. This obviously depends of the function ... — Matti P., Apr 10 '18 at 10:09
Because the last point involves logarithms, perhaps there is a mistake on the question: Is it really $x^2$ and not $2^x$? — Matti P., Apr 10 '18 at 10:12

score 1 · Answer 1 · edited Apr 10 '18 at 13:58

First of all, in order for an inverse function to be defined you must find if the initial function is 1-1, meaning that $f(x_1)=f(x_2)$ implies $x_1=x_2.$ Once you've checked that you take into account this: The main function and the inverse function happen to have the following property (The proof is very obvious if you consider the meaning of the inverse function): Their axis of symmetry is the line $y=x$. So when trying to sketch its graph you just find the symmetrical line with regard to $f(x)$ having as axis of symmetry the straight line $y=x$.

How to sketch the graph of an inverse function?

1 Answers1