In Exercises 31 to 48 , find f^-1(x). State any restrictions on the domain of f^-1(x). f(x)=x^2+

In Exercises 31 to 48 , find f^-1(x). State any restrictions...

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Key Concepts

Quadratic Functions

Quadratic functions, typically expressed in the form ax² + bx + c, have parabolic graphs and are not one-to-one when considered over all real numbers. To obtain an inverse for a quadratic function, the domain must be restricted to a monotonic interval, and techniques like completing the square may be used to facilitate solving for the variable when switching the roles of x and y.

Domain and Range Restrictions

Restricting the domain of a function is crucial when finding inverses, particularly for functions that are not naturally one-to-one. The domain restrictions ensure the existence of a unique inverse by limiting the original function to an interval where it is either entirely increasing or decreasing. The range of the original function then becomes the domain of its inverse.

Inverse Functions

Inverse functions effectively reverse the action of the original function by swapping the roles of inputs and outputs and solving for the new output variable. This process usually requires ensuring that the function is one-to-one over its domain, which may involve restricting the domain of the original function so that every input corresponds to a unique output.

Transcript

00:01 The first thing we're being asked to do in this problem is to find the inverse of f of x.

00:04 So in order to do this, we have to switch x with y and y with x.

00:08 Remember, f of x is equal to y.

00:10 So that means we'll have x equal to y squared plus 4y minus 1.

00:16 So now we need to solve for y, and i'm going to do this by using the completing the square method.

00:22 So i'm going to add one to both sides.

00:24 So we'll have x plus 1 equal to y squared plus 4y.

00:29 Now we have to find the constant that makes the perfect square trinomial.

00:33 So to do this, we do b over 2 square, where b is the coefficient of our middle term.

00:38 Well, 4 over 2 is 2, and 2 squared is 4.

00:43 But if i add 4 to the right hand side, we better add 4 to the left hand side.

00:47 So on the left, that will leave us with x plus 5.

00:50 And on the right, remember, it's a perfect square trinomial, so it will factor to one binomial being square.

00:56 Where the first term is the square of the first term, which is y.

01:00 The second term is the square of the last term, which is two.

01:04 And the sign is the sign of our middle term, which is plus.

01:08 So now, to solve for y, we're going to take the square root of both sides.

01:12 Don't forget, when you do that, you're going to have plus or minus...

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