Unlocking the Inverse of Quadratic Function: The Invertible Journey

	Definition of First Function	Substitute Second Function	Simplify
$f (f^{- 1} (x)) = ? x$	$2 f^{- 1} (x) - 3 = ? x$	$2 (\frac{x + 3}{2}) - 3 = ? x$	$x = x ✓$
$f^{- 1} (f (x)) = ? x$	$\frac{f ( x ) + 3}{2} = ? x$	$\frac{2 x - 3 + 3}{2} = ? x$	$x = x ✓$

Definition of First Function

Substitute Second Function

Simplify

f (f^{- 1} (x)) = ? x

2 f^{- 1} (x) - 3 = ? x

2 (\frac{x + 3}{2}) - 3 = ? x

x = x ✓

f^{- 1} (f (x)) = ? x

\frac{f ( x ) + 3}{2} = ? x

\frac{2 x - 3 + 3}{2} = ? x

x = x ✓

Domain of $f (x)$	Inverse Function
$0 \leq x < \infty$	$f^{- 1} (x) = x$
$- \infty < x \leq 0$	$f^{- 1} (x) = - x$

Domain of

f (x)

Inverse Function

0 \leq x < \infty

f^{- 1} (x) = x

- \infty < x \leq 0

f^{- 1} (x) = - x

Describe the error in finding and graphing the inverse of the function $f (x) = x + 4 .$

Let's take a look at the given exercise, paying close attention to each step, to see if we can find the error.

As can be seen in the steps the student wrote, the student began by attempting to find the inverse of y=sqrt(x+4) and its graph. Let's mimic this step by also solving for the inverse of the function. To do so, we will start by switching x and y. y=sqrt(x+4) ⇒ x=sqrt(y+4) So far, the notes are correct. Let's continue checking the steps and solve for y.

Because the range of f is y≥ 0, the domain of the inverse must be restricted to x≥0 as well. This is the mistake — the student forgot to restrict the domain of the inverse. g(x)=x^2-4, where x≥0 We will graph the inverse in a correct way.

A kite is flying at a height of

60

feet. The height

h

(in feet) of the kite above ground can be modeled by the following function.

h = - 4 t^{2} + 60

In this function rule,

t

is the time (in seconds) since the kite starts to fall.

Solve the equation for

t .

After how many seconds will the kite be

40

feet above the ground? Round your answer to one decimal place.

Consider the given equation for the height h above ground (in feet) of a kite with respect to the time since the kite starts to fall. h=-4t^2+60 We are asked to solve this equation for t. Let's do it!

Since a negative time does not make sense in the context of the problem, we will consider the positive t as the solution. t=sqrt(60-h)/2

We are also asked to find at what time the kite would be 40 feet above ground. We can do this by substituting this value for h into our formula and simplifying.

We can use a calculator to find this value rounded to one decimal place.

Therefore, the kite will be at 40 feet above ground about 2.2 seconds after it starts to fall.

Describe the change you could make to the function

f (x) = x^{2} + 3

so that its inverse is a function.

Let's start by drawing the graph of the given function. To do so, we will first make a table of values.

x	x^2+3	f(x)=x^2+3
- 2	( - 2)^2+3	7
- 1	( - 1)^2+3	4
0	0^2+3	3
1	1^2+3	4
2	2^2+3	7

Let's now plot and connect the obtained points. Keep in mind that the given function is quadratic and, therefore, its graph is a parabola.

Let's now apply the Horizontal Line Test.

Since the graph does not pass the Horizontal Line Test, the inverse of f is not a function. However, if we restrict the domain of f to be all non-negative real numbers, its graph will pass the Horizontal Line Test.

Great! Since the obtained graph passes the Horizontal Line Test, we can conclude that restricting the domain of f to be all non-negative real numbers makes its inverse a function. Please note that there are more ways to restrict the domain to make f invertible.

Inverses of Quadratic Functions

Catch-Up and Review

Are Quadratic and Square Root Functions Related?

Inverse Functions

Example

Invertible Function

Condition for Invertibility of a Function

Proof

Restricting the Domain of a Quadratic Function to Guarantee Invertibility

Hint

Solution

Identifying Invertible Functions

Finding the Inverse of a Quadratic Function

Modeling Falling Objects

Answer

Hint

Solution

Practice Finding a Point on the Inverse Function

Analyzing and Predicting the Trajectory of a Projectile

Hint

Solution

The Relation Between Quadratic and Radical Functions

	11 Theory slides
	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions