Question

Which of the following is true for the relation f(x) = 2x2 + 1? Only the inverse is a function. Only the equation is a function. Both the equation and its inverse are functions. Neither the equation nor its inverse is a function.

Answer 1

Both the equation and its inverse are functions.

In order to tell this, we first need to look at the inverse of the function. You can find the inverse of any function by switching the d(x) value and the x value. Then solve for the new d(x) value. The result will be your inverse function. Below is the step-by-step process for solving this one.

f(x) = 2x^2 + 1 ----> Switch the f(x) and x values

x = 2f(x)^2 + 1 ----> Subtract 1 from both sides

x - 1 = 2f(x)^2 ----> Divide both sides by 2

`(x - 1)/(2)` = f(x)^2 ----> Take the square root of both sides.

`\sqrt{(x - 1)/(2)}` = f(x) ----> Change the order for formatting purposes.

f(x) =
`\sqrt{(x - 1)/(2)}`

Now you have the inverse function. You'll notice with both cases, there is only one output for each input. No matter what we put in for x in both cases, there will be only one f(x) value that comes out. This is the definition of a function and thus proves both the original and new inverse are both functions.