Sign In
| 11 Theory slides |
| 8 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Every function has an inverse relation. If this inverse relation is also a function, then it is called an inverse function. In other words, the inverse of a function f is another function f^(- 1) such that they undo each other.
f(f^(- 1)(x))=x and f^(- 1)(f(x))=x
Therefore, f and f^(-1) are inverses of each other. Also, if x is the input of a function f and y its corresponding output, then y is the input of f^(- 1) and x its corresponding output.
f(x)=y ⇔ f^(- 1)(y)=x
Consider a function f and its inverse f^(- 1). f(x)=2x-3 and f^(-1)(x)= x+32
f^(- 1)(x)= x+3/2
2 * a/2= a
Remove parentheses
Subtract term
Definition of First Function | Substitute Second Function | Simplify | |
---|---|---|---|
f(f^(- 1)(x))? =x | 2f^(- 1)(x)-3? =x | 2( x+3/2)-3? =x | x=x ✓ |
f^(- 1)(f(x))? =x | f(x)+3/2? =x | 2x-3+3/2? =x | x=x ✓ |
Therefore, f and f^(- 1) undo each other. The graphs of these functions are each other's reflection across the line y=x. This means that the points on the graph of f^(- 1) are the reversed points on the graph of f.
The characteristic feature of a quadratic function is that the function rule involves raising the input value to the second power. This feature is especially clear for the quadratic parent function. f(x) = x^2 The inverse operation of raising a value to the second power is taking its square root. In other words, squaring a number and calculating its square root are operations that undo each other. The following example demonstrates this concept. 8^2 = 64 sqrt(64)=8 For this reason, square root functions can be used to define the inverse of quadratic functions. There is something else to consider, however. There is another possibility for the previous example. (- 8)^2 = 64 sqrt(64)=8 The fact that the square of two different numbers can have the same value complicates defining an inverse for a quadratic function. In this case, the existence of an inverse operation is not enough for a function to be invertible. Before exploring the conditions a function must meet to be invertible, the definition of an invertible function will be given.
A function is invertible if its inverse relation is also a function. For example, consider the following linear function. f(x) =0.5x-2 The graph of its inverse relation can be found by reflecting the graph of f(x) across the line y=x.
There is a condition that a function must satisfy in order to be invertible.
This condition implies that if a function fails the Horizontal Line Test, then it is not invertible. However, some functions do not satisfy this condition for invertibility but can become invertible by restricting their domain. For example, consider the quadratic parent function f(x)=x^2.
This function has different x-values where the same y-value is assigned. The foll how the function fails the Horizontal Line Test.
Now, the graph of f(x) will be reflected across the line y=x to visualize its inverse relation.
The inverse is not a function because it has input values with two different outputs assigned. The following graph further confirms this relation by showing how it fails the Vertical Line Test.
Therefore, f(x)=x^2 is not invertible because its inverse is not a function. Now, consider if the domain of f(x) is restricted to x≥ 0. In that case, every x-value gets assigned a unique y-value. With the domain of f(x) defined this way, f^(- 1)(x)=sqrt(x) is a valid inverse function.
Consider a function f(x) that assigns the same output to multiple inputs.
Heichi is so fascinated about the invertibility of quadratic functions, that he started a study group.
He brought the following function rule.
f(x) = 1/2(x-2)^2+8 Heichi asks his friends to help him identify in which intervals the function is invertible. Select all the options that correspond to these intervals.The following applet shows different quadratic functions. The representation of these functions alternates between showing their graph and stating their function rule. Determine whether or not the given function is invertible in the indicated domain.
To begin, since f(x)=y describes the input-output relationship of the function, replace f(x) with y in the function rule. f(x)= 2x-1/3 → y= 2x-1/3
Because the inverse of a function reverses x and y, the variables can be switched. Notice that every other piece in the function rule remains the same. y= 2 x-1/3 switch x= 2 y-1/3
LHS * 3=RHS* 3
LHS+1=RHS+1
.LHS /2.=.RHS /2.
Rearrange equation
Just as f(x)=y shows the input-output relationship of f, so does f^(- 1)(x)=y. Therefore, replacing y with f^(- 1)(x) gives the rule for the inverse of f. y=3x+1/2 → f^(- 1)(x)=3x+1/2 Notice that in f, the input is multiplied by 2, decreased by 1, and divided by 3. From the rule of f^(- 1), it can be seen that x undergoes the inverse of these operations in the reverse order. Specifically, x is multiplied by 3, increased by 1, and divided by 2.
LHS-k=RHS-k
.LHS /a.=.RHS /a.
sqrt(LHS)=sqrt(RHS)
LHS+h=RHS+h
Rearrange equation
Rewrite y as f^(- 1)(x)
Domain: 0 ≤ x ≤ 3
Range: 0 ≤ y ≤ 3
However, all the x-values in the interval 0 ≤ x ≤ 3 have unique y-values associated. This is why restricting the function's domain to the interval 0 ≤ x ≤ 3 makes the function invertible.
f(x)= - 5x^2+45 → y= - 5x^2+45
Now, the variables y and x will be switched because the inverse of a function reverses inputs and outputs.LHS-45=RHS-45
.LHS /(-5).=.RHS /(-5).
Put minus sign in numerator
Distribute - 1
Commutative Property of Addition
sqrt(LHS)=sqrt(RHS)
Rearrange equation
x= 30
Subtract term
Calculate quotient
Calculate root
Round to 1 decimal place(s)
x= 20
Subtract term
Calculate quotient
Calculate root
Round to 1 decimal place(s)
The following applet shows a quadratic function and a specific point on its graph. The domain of the given function is restricted and the function is invertible. Find the coordinates of the corresponding point on the graph of its inverse function. Write the answer rounding to two decimal places.
a = a+ 6400- 6400
Commutative Property of Addition
Subtract term
a^2± 2ab+b^2=(a± b)^2
cc Quatratic Function& Vertex f(x) = a(x- h)^2+ k & ( h, k) [1em] Quatratic Function & Vertex f(x) = - 1/160(x- 80)^2+ 90 & ( 80, 90) Note that x represents horizontal displacement and f(x) represents height. That means in this situation they both only make sense if they are non-negative. x≥ 0 and f(x)≥ 0 The following graph of f(x)=x- x^2160+50 is shown bounded with these restrictions.
f(x)= - 1/160(x-80)^2+90 ↓ y= - 1/160(x-80)^2+90
Next, the variables y and x will be switched because the inverse of a function reverses inputs and outputs.LHS-90=RHS-90
LHS * (- 160)=RHS* (- 160)
Distribute - 160
Commutative Property of Addition
sqrt(LHS)=sqrt(RHS)
LHS+80=RHS+80
Rearrange equation
x= 3
Domain of f(x) | Inverse Function |
---|---|
0 ≤ x < ∞ | f^(- 1)(x)=sqrt(x) |
- ∞ < x ≤ 0 | f^(- 1)(x)=- sqrt(x) |
A great method to explore this is by using the following interactive graph. The graph gives various choices of domains for f(x) = x^2. The function's corresponding inverse relation is shown by reflecting f(x) across y=x.
Consider the following quadratic function. f(x)=4x^2-9
We want to solve the equation y=f(x) for x. Let's consider the given function rule. f(x)=4x^2-9 We will now substitute the given expression for f(x) in y=f(x) and then solve for x.
Since we are asked to find the positive solution, the answer is x=sqrt(y+94) or x= sqrt(y+9)2.
Note that the input is represented by the x-variable, and the output is represented by the y-variable. As we found in Part A, two inputs — one positive and one negative — can give us one output. We are asked to find the negative input when the output is 7. To do so, we will use the equation that gives the negative x-values. x=-sqrt(y+9)/2 Now, we will substitute 7 for y into the above equation.
Consider the following graph.
Which graph represents the reflection of f across the line y=x?
We will reflect the graph of f across the line y=x. Let's start by drawing y=x. Since it is already in the slope-intercept form, we can easily identify its slope and y-intercept. y=x ⇔ y= 1x+ 0 Our line will pass through the origin with a slope of 1.
Next, we will reflect the points of the given function across y=x. Remember that when a point (x,y) is reflected across the line y=x, the coordinates of the reflected point will be (y,x). With this in mind, let's make a table to identify the coordinates of five reflected points.
Given Points | Reflected Points |
---|---|
(-2,6) | (6,-2) |
(-1,0) | (0,-1) |
(0,-2) | (-2,0) |
(1,0) | (0,1) |
(2,6) | (6,2) |
Let's now connect the reflected points in a smooth curve.
By removing the other elements, let's focus solely on the reflected graph g.
According to what we graphed, the correct answer is option D.
Consider the following quadratic function. f(x)=1/2x^2-6
We want to find an interval on which we can define the inverse of the given function. Since the function is quadratic, we have to restrict the domain to avoid having repeated output values. To do so, let's identify the vertex of the function. cc Quadratic Function& Vertex f(x) = a(x- h)^2+ k & ( h, k) [1em] Quadratic Function & Vertex f(x) = 1/2(x- 0)^2+( -6) & ( 0, -6) We can restrict the domain to values either equal or less than or equal or greater than the x-coordinate of the vertex. x ≤ 0 or 0 ≤ x Notice that any smaller interval included in the above intervals can be considered a solution. However, in the domains that include both positive and negative numbers, our function is not invertible. Therefore, the answer is option B.
We will now find the inverse of our function. To do so, we will first replace f(x) with y.
f(x)=1/2x^2-6 ⇔ y=1/2x^2-6
To algebraically determine the inverse of the given relation, we exchange x and y and solve for y.
ccc
Given Equation& & Inverse Equation
y=1/2 x^2-6 & & x=1/2 y^2-6
The result of isolating y in the new equation will be the inverse of the given function.
Let's now use function notation and replace y with g(x). y=± sqrt(2x+12) ⇕ g(x)=± sqrt(2x+12) Notice that we ended with two expressions — one non-negative and one non-positive. Since the domain of f(x) chosen in Part A is 0≤ x, all its input values are non-negative. Inputs of f(x) are the outputs of its inverse g(x). Therefore, the inverse function's rule should be the non-negative expression. g(x)=sqrt(2x+12)
Consider the following graph.
We are given the following function and its graph. We want to determine whether the inverse of this function is also a function or not.
We will use the Horizontal Line Test. If the horizontal lines intersect the graph once, then the inverse is also a function. Conversely, if there is even one horizontal line that intersects the graph more than once, then the inverse is not a function. With this in mind, we will draw several horizontal lines across the given graph of a function.
We can see above that there are horizontal lines that intersect the curve at more than one point. Therefore, the inverse of the given function is not a function.