Inverse Functions

Test your understanding of when inverse functions exist, how to compute them algebraically, their domain/range relationships, graphical symmetry, and the difference between inverse notation and reciprocal

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Views #: 316

Questions #: 20

Time: 19 minutes

Pass Score: 80.0%

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When Does an Inverse Function Exist?

1 pts

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The diagram below shows a function \(f\) and its reverse correspondence \(f^{-1}\), reading arrows both ways.

Under which condition does a function \(f: A \to B\) have an inverse function \(f^{-1}\)?

When f is surjective (onto) only

When f is injective (one-one) only

When f is bijective (both one-one and onto)

Always — every function has an inverse

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Domain and Range of the Inverse Function

2 pts

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If \(f: A \to B\) is bijective and \(f^{-1}\) is its inverse, complete the two statements: the domain of \(f^{-1}\) is (1), and the range of \(f^{-1}\) is (2).

Please drag and drop the selected option in the right place or type it instead

the range of f

the domain of f

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Explanation

The Inverse Undoes the Original Function

1 pts

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If \(f(x) = y\), which two identities hold for any \(x\) in the domain of \(f\) and any \(y\) in the range of \(f\)?

\(f^{-1}(f(x)) = x\) and \(f(f^{-1}(y)) = y\)

\(f^{-1}(f(x)) = y\) and \(f(f^{-1}(y)) = x\)

\(f^{-1}(f(x)) = 0\) and \(f(f^{-1}(y)) = 1\)

\(f^{-1}(x) = f(x)\) and \(f(y) = f^{-1}(y)\)

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Explanation

f⁻¹ Does NOT Mean 1/f(x)

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True or False: The notation \(f^{-1}(x)\) means the same as \(\frac{1}{f(x)}\).

True

False

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Is the Inverse Function Unique?

1 pts

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If a function \(f\) has an inverse, how many inverse functions does it have?

None — inverse functions do not exist for real functions

Exactly one — the inverse is uniquely determined

Two — one for each direction

Infinitely many — one for each element of the codomain

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Is f⁻¹ Also Bijective?

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If \(f: A \to B\) is bijective and \(f^{-1}: B \to A\) is its inverse, what type of function is \(f^{-1}\)?

Only injective

Only surjective

Also bijective (one-one and onto)

Many-one

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Explanation

Three-Step Process: Sort the Steps

2 pts

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Please drag and drop the options to sort them

To find a formula for \(f^{-1}(x)\) from \(y = f(x)\) (for simple functions), there is a three-step process. Sort the steps into the correct order.

Replace y by x to write the inverse in the form y = f⁻¹(x)

Solve the equation y = f(x) for x in terms of y

Name the resulting expression in y as f⁻¹(y)

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Find the Inverse of f(x) = 3x − 2

1 pts

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Using the three-step process, find \(f^{-1}(x)\) for \(f(x) = 3x - 2\), \(x \in \mathbb{R}\).

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

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

\(f^{-1}(x) = \dfrac{x + 2}{3}\)

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

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Explanation

Find the Inverse of f(x) = x/(1−x)

2 pts

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Find \(f^{-1}(x)\) for \(f(x) = \dfrac{x}{1 - x}\).

\(f^{-1}(x) = \dfrac{x}{1 + x},\; x \ne -1\)

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

\(f^{-1}(x) = \dfrac{1}{x},\; x \ne 0\)

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

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Find the Inverse of f(x) = 2x

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Enter the formula for \(f^{-1}(x)\) if \(f(x) = 2x\).

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Explanation

Graph Symmetry of Inverse Functions

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The diagram below illustrates the geometric relationship between the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\).

With respect to what line are the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) symmetric?

The x-axis

The y-axis

The line \(y = x\)

The line \(y = -x\)

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Mutually Inverse Functions: y = x³ and y = ∛x

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Are \(y = x^3\) and \(y = \sqrt[3]{x}\) mutually inverse functions?

Yes

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Explanation

Inverse of the Exponential Function

2 pts

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The exponential function \(y = e^x\) defines a bijective mapping from \((-\infty, \infty)\) onto \((0, \infty)\). What is its inverse function, and what are the inverse's domain and range?

Inverse: \(y = e^{-x}\); domain \((0, \infty)\); range \((-\infty, \infty)\)

Inverse: \(y = \ln x\); domain \((0, \infty)\); range \((-\infty, \infty)\)

Inverse: \(y = \ln x\); domain \((-\infty, \infty)\); range \((0, \infty)\)

Inverse: \(y = \log_{10} x\); domain \((-\infty, \infty)\); range \((0, \infty)\)

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Explanation

Match Functions to Their Inverses

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To complete the line match

Click on an item in the first group
Click on an item in the second group
To delete a match, double click on a line

Function f(x)

\(2x\)

\(x^3\)

\(3x - 2\)

\(10^x\)

Inverse f⁻¹(x)

\(\log_{10} x\)

\(\dfrac{x}{2}\)

\(\sqrt[3]{x}\)

\(\dfrac{x + 2}{3}\)

Match each function to its inverse.

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Can We Always Find a Formula for f⁻¹?

1 pts

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Consider \(f(x) = x^5 + 2x + 1\). Can we find an explicit formula for \(f^{-1}(x)\) by solving for \(x\)?

Yes — all polynomial inverses have explicit formulas

No — general polynomial equations of degree ≥ 5 cannot be solved in terms of their coefficients

Yes — but only if the function is bijective

No — because x⁵ + 2x + 1 is not bijective

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Explanation

Mutual Inverse Functions Share One Curve

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The functions \(y = f(x)\) and \(x = f^{-1}(y)\) — viewed in the same xy-plane — describe how many distinct curves?

Two distinct curves — one for each function

One and the same curve in the xy-plane

Three curves — the original, the inverse, and their intersection

None — they cancel each other out

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Which Functions Have an Inverse? (Multiple Answers)

2 pts

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Which of the following functions (from \(\mathbb{R}\) to \(\mathbb{R}\)) have an inverse? (Select all that apply.)

\(f(x) = x^3\)

\(f(x) = x^2\)

\(f(x) = 2x + 1\)

\(f(x) = \sin x\)

\(f(x) = e^x\) from \(\mathbb{R}\) onto \((0,\infty)\)

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Classify These Functions: Has Inverse or No Inverse

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Function

×f(x) = 3x - 2

×f(x) = |x|

×f(x) = x/(1-x)

×f(x) = cos x

Has Inverse

No Inverse

Classify each function below as Has Inverse or No Inverse (as functions from \(\mathbb{R}\) to \(\mathbb{R}\) unless otherwise noted).

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Explanation

The Axis Convention for Inverse Functions

1 pts

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To draw the graph of \(f^{-1}\) with \(x\) as the independent variable (horizontal axis), what must be done to the formula \(x = f^{-1}(y)\)?

Leave it as is — x is already on the horizontal axis

Replace y by x so that f⁻¹ is expressed as y = f⁻¹(x)

Swap the domain and codomain labels only

Plot x on the vertical axis and y on the horizontal axis

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Explanation

Domain Restriction for Inverse

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Consider the function \( f(x)=x^2 \) defined on the domain \( x \in \mathbb{R} \). Does \( f(x) \) have an inverse function on this domain?

Yes, \(f^{-1}(x) = \sqrt{x}\)

No, because the function is not one-one (injective) on \(\mathbb{R}\)

Yes, because \(f(x)\) is onto (surjective)

No, because \(f(x)\) is not a function

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Explanation

Year 10

When Does an Inverse Function Exist?

Correct Answer

Explanation

Domain and Range of the Inverse Function

Correct Answer

Explanation

The Inverse Undoes the Original Function

Correct Answer

Explanation

f⁻¹ Does NOT Mean 1/f(x)

Correct Answer

Explanation

Is the Inverse Function Unique?

Correct Answer

Explanation

Is f⁻¹ Also Bijective?

Correct Answer

Explanation

Three-Step Process: Sort the Steps

Correct Answer

Explanation

Find the Inverse of f(x) = 3x − 2

Correct Answer

Explanation

Find the Inverse of f(x) = x/(1−x)

Correct Answer

Explanation

Find the Inverse of f(x) = 2x

Correct Answer

Explanation

Graph Symmetry of Inverse Functions

Correct Answer

Explanation

Mutually Inverse Functions: y = x³ and y = ∛x

Correct Answer

Explanation

Inverse of the Exponential Function

Correct Answer

Explanation

Match Functions to Their Inverses

Function f(x)

Inverse f⁻¹(x)

Correct Answer

Explanation

Can We Always Find a Formula for f⁻¹?

Correct Answer

Explanation

Mutual Inverse Functions Share One Curve

Correct Answer

Explanation

Which Functions Have an Inverse? (Multiple Answers)

Correct Answer

Explanation

Classify These Functions: Has Inverse or No Inverse

Correct Answer

Explanation

The Axis Convention for Inverse Functions

Correct Answer

Explanation

Domain Restriction for Inverse

Correct Answer

Explanation

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