Question

In Exercises \(7-12,\) determine whether the function has an inverse function. $$y=x^{2}+5 x$$

Short answer

The function \(y = x^2 + 5x\) does have an inverse, because it is a one-to-one function.

Step-by-step solution

Find the derivative

The derivative of the function will help to identify the increasing and decreasing intervals of the function, which is crucial in figuring out if it's one-to-one. The derivative of \(y = x^2 + 5x\) is \(y' = 2x + 5\).

Analyze the derivative

Set \(y' = 0\) and solve for \(x\). Using \(2x + 5 = 0\), x=-5/2. This is the x-coordinate where the function changes from increasing to decreasing or vice versa. For \(x < -5/2\), the function decreases, and for \(x > -5/2\), the function increases. Since it doesn't decrease and increase simultaneously in any interval, we conclude that it only exists once in every value of \(y\).

Apply the horizontal line test

The horizontal line test is a theoretical test that involves imagining drawing a horizontal line through the graph of the function. If at any point the line crosses the graph more than once, then the function does not have an inverse. For this function, any horizontal line will only intersect the graph once - so the function is one-to-one and thus does have an inverse.