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Discussion
Beth want to figure out how much water evaporates from the lake near her house throughout the day. She has expressions modeling the temperature T (°C) as a function of time t (hours after midnight) and evaporation E (liters/hour) as a function of temperature. Explain how Beth could relate the time of day to the evaporation rate of the lake.
Beth has two formulas: one giving temperature in terms of time,
and one giving evaporation rate in terms of temperature,
To get evaporation directly as a function of the time t, she proceeds in two steps:
For a chosen t, compute the temperature T.
Substitute that temperature into the evaporation formula to get E.
In other words, “time → temperature → evaporation.” The result is a single formula
which for any hour t after midnight tells her the evaporation rate at that time.
Composite functions
Functions can be composed by passing the output of one into the other. We use the symbol ∘, and pay close attention to the order in which functions are composed:
To find an expression for f(g(x)), substitute g(x) for x in the expression for f(x).
Example
Given f(x)=x2 and g(x)=2x−1, find (a) (f∘g)(x) and (b) (g∘f)(x).
Exercise
Let f(x)=5x2+7 and g(x)=−x−2.
Find
(f∘g)(x)
(g∘f)(x)
Select the correct option
Discussion
If f−1, the inverse of f, is the function that "undoes" f and it is given that f(a)=b, find f−1(b).
The inverse function undoes whatever f does: if y=f(x), then f−1(y)=x. Substituting a and b for x and y yields: if b=f(a), then f−1(b)=a.
Since we are given that f(a)=b, we must have
Finding inverse of specific value
We can find x=f−1(b) by applying the function to both sides:
So finding f−1(b) is equivalent to solving f(x)=b.
Graphically, find f−1(b) is equivalent to being given y=b, and finding the value of x for which that is true:
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Example
We can also solve for the inverse of a point algebraically. Given f(x)=3x+2, find f−1(11).
so f−1(11)=3.
Exercise
Consider the function f(x)=x3−5.
Find f−1(3).
Select the correct option
Discussion
Recalling that f(a)=b and f−1(b)=a are equivalent, if (a,b) is a point on f, find a point on f−1.
Since (a,b) lies on f, we have
By the recalled equivalence f(a)=b⟺f−1(b)=a, it follows that
so the point (b,a) lies on f−1.
Graphs of inverse functions
The graph of a function f shows all the points (x,f(x)). Since f−1 undoes f, its graph will show all the points (f(x),x). In other words, the x and y values are swapped.
This is equivalent to reflecting the curve y=f(x) in the line y=x:
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Discussion
How are the domain and range of a function connected to the domain and range of its inverse?
If f is a function with domain D and range R, then by the definition of an inverse function:
Here, x must be in the domain D of f, and y must be in the range R of f.
When considering the inverse function f−1, the input is y (from the range of f), and the output is x (from the domain of f). Therefore:
The domain of f−1 is the range R of f.
The range of f−1 is the domain D of f.
In summary:
Example:
Let f be defined by f(x)=√x.
The domain of f is [0,∞).
The range of f is [0,∞).
The inverse function is f−1(y)=y2, which has:
Domain [0,∞) (the original range of f)
Range [0,∞) (the original domain of f)
Domain & range of inverse functions
Since f−1 undoes f, the domain of f−1 is all the possible values f could output. That is, the domain of f−1 is the range of f.
The range of f−1 is all the possible values that could have gone into f. Thus, the range of f−1 is the domain of f.
Exercise
Consider the function f(x)=4−√x+2.
For the function f−1, find
the domain,
the range.
Select the correct option
Discussion
Find (f−1∘f)(x).
Let x be any element in the domain of f, and set
so that by definition f(x)=y. Then
Inverse applied to function is identity x
Formally, the inverse function is such that
We call x the identity function, as I(x)=x composed with any function gives the same function.
Checkpoint
Find (f∘f−1)(3).
Select the correct option
Discussion
It is given that y=f(x), explain how you would solve for x=f−1(y).
Hence, given f(x)=x+2x−1, solve for an expression of f−1(y).
Part (a)
Here’s a concrete example, followed by the general process.
Example
Take f(x)=2x+1. We start with
Hence
We confirm that this is the correct inverse:
If you prefer to write the inverse as a function of x, simply swap variables at the end:
General steps
Write the definition of f:
Solve this equation algebraically for x, undoing the operations in reverse order.
The result x=⋯ is precisely f−1(y).
(Optional) If you want the inverse in the form f−1(x), swap the letters x and y.
Part (b)
We set
and solve for x in terms of y.
Hence
If you prefer the inverse as a function of x, simply rename y→x:
Finding inverse functions
Inverse functions f−1(x) can be found algebraically by switching x and y in the expression for f(x) and attempting to isolate y.
Exercise
Consider f(x)=x−13x−2.
Find f−1(x).
Select the correct option
Discussion
Not all functions are invertible (have an inverse). Recalling what separates functions from other relations, how could we test if a function has an inverse?
A relation is a function precisely when each input x yields exactly one output y. Graphically, that means no vertical line meets its graph more than once (the “vertical‐line test”).
For an inverse to be a function we must have that each original output y comes from exactly one x. Equivalently, if we swap x and y, the new graph must pass the vertical‐line test. But swapping axes turns vertical lines into horizontal lines.
Hence:
• Algebraic test: whenever
we must have x1=x2.
• Graphical test: no horizontal line meets the graph of f more than once.
If either holds, f is one‐to‐one and so admits an inverse that is itself a function.
Existence of inverse function
Recall that a function f must pass the vertical line test to guarantee that each input gives at most one output.
Since f−1 is also a function, it too must pass the vertical line test. But since the graph of f−1(x) is a reflection of the graph of f(x) in the line y=x, the graph of f(x) must then pass the horizontal line test:
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A function that passes the horizontal line test is said to be one to one - each input yields exactly one distinct output. Such a function is said to be invertible, which means f−1 exists.
Exercise
For which whole numbers n is f(x)=xn invertible?
Select the correct option
Discussion
It is given that f(x)=b−x. Find f−1(x).
Set y=b−x. Then solve for x:
Swapping x and y to obtain the inverse gives
Notice, f(x)=f−1(x)=b−x.
Self-inverse functions
A function is said to be self-inverse if it is its own inverse:
Since the graph of f−1(x) is the mirror image of f(x) in the line y=x, the graph of a self-inverse function must be symmetric in the line y=x.
Discussion
You may notice that f(x)=x2 is not invertible over its whole domain. However, we consider the square root to be the inverse operation of the square. Find the inverse function of g(x)=√x.
Let y=g(x)=√x. Then
Swapping x and y to write the inverse gives
Hence
Since g(x)=√x only outputs non-negative values, its range is [0,∞[. That becomes the domain of g−1, so
That is, we have found that x2 is indeed the inverse of √x, but only after we limit its domain to all positive numbers!
Finding inverse function with domain restriction
When a function f is not one to one, ie it does not pass the horizontal line test, its domain needs to be restricted for the inverse to exist.
Exercise
The function f(x)=4x2−4x+1 is defined on the domain x≤a.
Find the largest possible value of a such that f−1 exists.
Select the correct option
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