Question

How to find the inverse of $5x - 3$?

Answer 1

Hint: In order to find the inverse of the given linear equation, we need to let our equation be $y = 5x - 3$. Then we need to exchange the variables $x$ and $y$ with each other. We further solve the sum and isolate $y$ to get our new inverse function in terms of $x$.

Complete step-by-step solution:
In the given question, we are asked to find the inverse of the given function.
Thus, let the function be $f\left( x \right) = 5x - 3$.
Now in order to find the inverse function, we need to take this function as $y = 5x - 3$.
Let us exchange the variables $x$ and $y$ as the first step to find the inverse.
$ \Rightarrow x = 5y - 3$
Let us add $ + 3$ to both sides of the equation. Thus, we have:
$ \Rightarrow x + 3 = 5y$
On dividing both sides with $5$, we get:
$ \Rightarrow y = \dfrac{{x + 3}}{5}$
We rearrange it to get:
$ \Rightarrow y = \dfrac{x}{5} + \dfrac{3}{5}$

Thus our inverse function is ${f^{ - 1}}\left( x \right) = \dfrac{x}{5} + \dfrac{3}{5}$

Note: Finding the inverse function of a linear equation is relatively easier than finding the inverse of , say a quadratic equation. This is because the domain and range of linear equations span over all the real numbers, given that the domain is not restricted. When we find the inverse of a function of a linear equation, the domain and range get swapped with each other.
 All quadratic functions may not have an inverse. We can find the inverse when the domain and the range are well defined. The first thing to notice while finding the inverse of a quadratic equation is the coefficient of $a$ in$a{x^2} + bx + c$. If we plot the inverse function on a graph, then we get a parabola. This parabola has the following conditions:
If $a > 0$, then the equation defines a parabola whose ends point upwards.
If $a < 0$, then the equation defines a parabola whose ends point downwards.
$a \ne 0$, since then the function will become linear and not quadratic.