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Quizzes > High School Quizzes > Mathematics

6.04 Inverses Practice Quiz

Master inverse functions with practical exercises

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Paper art promoting the Inverse Insights quiz for high school algebra and precalculus students

What is the definition of an inverse function?
A function with the same domain as the original
A function that reverses the operations of the original function
A function that is always increasing
A function that has no inputs
The inverse function undoes what the original function does; f(f❻¹(x)) = x. This definition captures the concept of reversing the operations.
If f(x) = 3x + 2, what is the inverse function f❻¹(x)?
f❻¹(x) = 3x - 2
f❻¹(x) = (x - 2)/3
f❻¹(x) = 3/(x - 2)
f❻¹(x) = (x + 2)/3
To find the inverse, you swap x and y and solve for y which gives y = (x - 2)/3. This process correctly reverses the operations of f(x).
Which property must a function have to ensure it has an inverse function?
It must be symmetric about the y-axis
It must pass the horizontal line test
It must be periodic
It must be a quadratic function
A function must be one-to-one, which is confirmed if it passes the horizontal line test. This property ensures that each output is unique and can be reversed.
What does the equation f(f❻¹(x)) = x represent in inverse functions?
It suggests that f has no inverse
It implies that the function f is quadratic
It indicates that f and f❻¹ are the same function
It shows that applying a function and its inverse returns the original input
This equation illustrates the defining property of inverse functions: applying the function and its inverse cancels each other out. It confirms that the original input is retrieved accurately.
For the function f(x) = x³, which of the following is the inverse function?
f❻¹(x) = 3√x
f❻¹(x) = x^(1/2)
f❻¹(x) = ∛x
f❻¹(x) = x²
The cube function is one-to-one and its inverse is the cube root function. This correctly returns the original input when composed with f(x).
Given the function f(x) = 2x - 4, find its inverse function f❻¹(x) in simplest form.
f❻¹(x) = 2/(x - 4)
f❻¹(x) = (x + 4)/2
f❻¹(x) = 2x + 4
f❻¹(x) = (x - 4)/2
To invert f(x) = 2x - 4, you solve y = 2x - 4, swap the variables, and then solve for y, resulting in f❻¹(x) = (x + 4)/2. This correctly isolates y in the inverse function.
If f(x) = (x - 3)², why is it necessary to restrict the domain for the inverse?
Because f(x) is not one-to-one over all real numbers
Because f(x) never touches the x-axis
Because f(x) is always increasing
Because f(x) is not a quadratic function
The function f(x) = (x - 3)² is a quadratic, which is not one-to-one without a restricted domain. Limiting the domain ensures that each output comes from a unique input, making the inverse well-defined.
For the function f(x) = (5x + 1)/2, what is the inverse function?
f❻¹(x) = (2x - 1)/5
f❻¹(x) = (5x + 1)/2
f❻¹(x) = (x - 1)/5
f❻¹(x) = (5x - 1)/2
By interchanging x and y in the equation f(x) = (5x + 1)/2 and solving for y, you find that f❻¹(x) = (2x - 1)/5. This process reverses the arithmetic operations applied in f(x).
If f(x) = 1/(x - 2), what is the inverse function f❻¹(x)?
f❻¹(x) = x + 2
f❻¹(x) = 1/(2 - x)
f❻¹(x) = 1/x + 2
f❻¹(x) = 2 - 1/x
Starting with f(x) = 1/(x - 2), swapping x and y gives x = 1/(y - 2). Solving for y leads to f❻¹(x) = 1/x + 2, which is the correct inverse form.
Which of the following statements best describes the relationship between a function and its inverse on a graph?
They are mirror images of each other across the line y = x
They are symmetric about the x-axis
They are identical if the function is linear
They are symmetric about the y-axis
The graph of a function and its inverse are reflections of each other across the line y = x. This symmetrical property is a key visual characteristic of inverse functions.
For f(x) = √(x + 7), what is its inverse function with proper domain restrictions?
f❻¹(x) = x² - 7, with x ≤ 0
f❻¹(x) = x² - 7, with x ≥ 0
f❻¹(x) = x² + 7, with x ≥ 7
f❻¹(x) = √(x - 7)
Since f(x) = √(x + 7), squaring both sides of the equation y = √(x + 7) leads to x + 7 = y², and solving for y gives f❻¹(x) = x² - 7. The restriction x ≥ 0 is applied because the square root function yields nonnegative outputs.
If f(x) is one-to-one and invertible, what is the result of f❻¹(f(x))?
x²
f❻¹(x)
f(x)
x
Composing a function with its inverse always returns the original input, as f❻¹(f(x)) = x. This is a fundamental property of invertible functions that confirms the reversal of operations.
What is the inverse function of f(x) = -4x + 9?
f❻¹(x) = (9 - x)/4
f❻¹(x) = 4x - 9
f❻¹(x) = (-9 - x)/4
f❻¹(x) = (x - 9)/-4
By swapping the variables in f(x) = -4x + 9 and solving for y, you get f❻¹(x) = (9 - x)/4. This correctly reverses the operations of the original function.
How would you verify algebraically that two functions, f(x) and g(x), are inverses of each other?
By checking if their graphs intersect
By showing that f(g(x)) = x and g(f(x)) = x
By verifying that f(x) = g(x)
By confirming they have the same slope
To verify that two functions are inverses, you must show that composing them in both orders yields the identity function. This confirms that each function reverses the effect of the other.
If f(x) = (x+2)/(x-3), what must be true for f(x) to be invertible?
The range of f(x) should include all real numbers
f(x) must be one-to-one, ensuring that each y-value comes from a unique x-value (with x 3)
f(x) must be symmetrical about the y-axis
f(x) must only have integer outputs
For a function to be invertible, it must be one-to-one so that each output corresponds to a unique input. In the case of f(x) = (x+2)/(x-3), excluding x = 3 is necessary to avoid undefined values and ensure one-to-one behavior.
Consider the function f(x) = (3x - 5)/(2x + 7). What is the inverse function f❻¹(x) in its simplified form?
f❻¹(x) = (-7x - 5)/(2x - 3)
f❻¹(x) = (7x + 5)/(2x - 3)
f❻¹(x) = (-7x + 5)/(2x - 3)
f❻¹(x) = (-5 - 7x)/(2x + 3)
By swapping x and y in f(x) = (3x - 5)/(2x + 7) and solving for y, the inverse is derived as f❻¹(x) = (-7x - 5)/(2x - 3) after proper simplification.
Determine the inverse of the function f(x) = 2√(x - 1) + 3.
f❻¹(x) = 2√(x - 3) + 3
f❻¹(x) = ((x - 3)²)/4 + 1
f❻¹(x) = ((x + 3)²)/4 + 1
f❻¹(x) = ((x - 3)²)/2 + 1
Subtract 3 and divide by 2 to isolate the square root, then square both sides to eliminate it, yielding f❻¹(x) = ((x - 3)²)/4 + 1. This series of steps correctly reverses the operations in f(x).
For the function f(x) = ln(x - 2), find the inverse function f❻¹(x).
f❻¹(x) = e^(x - 2)
f❻¹(x) = ln(x - 2)
f❻¹(x) = eˣ + 2
f❻¹(x) = ln(x + 2)
Starting with y = ln(x - 2) and swapping x and y, you exponentiate to get eˣ = y - 2 and then solve for y, resulting in f❻¹(x) = eˣ + 2, which correctly reverses the logarithmic operation.
If f(x) = x² + 1, can f have an inverse function? If not, what modification can be made?
No; quadratic functions never have inverses even with domain restrictions
Yes; the inverse is f❻¹(x) = x² - 1
Yes; its inverse is f❻¹(x) = √(x - 1)
No; restrict the domain to x ≥ 0 or x ≤ 0 to ensure one-to-one behavior
Quadratic functions like f(x) = x² + 1 are not one-to-one over their entire domain. Restricting the domain to either x ≥ 0 or x ≤ 0 creates a one-to-one function, allowing an inverse to exist.
Find the inverse of the function f(x) = (x - 4)/(x + 2) and state one restriction on its domain.
f❻¹(x) = (-2x - 4)/(x - 1), with x 1
f❻¹(x) = (2x + 4)/(x - 1)
f❻¹(x) = (2x - 4)/(x - 1)
f❻¹(x) = (-2x - 4)/(x + 1)
After swapping x and y in f(x) = (x - 4)/(x + 2) and solving for y, the inverse function is determined to be f❻¹(x) = (-2x - 4)/(x - 1). The restriction x 1 is necessary to avoid division by zero.
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Study Outcomes

  1. Analyze the concept of inverse functions and their properties.
  2. Evaluate conditions that determine when a function has an inverse.
  3. Calculate and simplify inverse functions using algebraic methods.
  4. Verify inverse relationships through graphical and analytical techniques.

6.04 Quiz: Inverses Cheat Sheet

  1. Inverse Function Basics - An inverse function essentially "undoes" what the original function does by swapping inputs and outputs. For instance, if f(x) = 2x + 3, then f❻¹(x) brings you back to x by reversing those steps. This is the cornerstone of algebraic operations! Inverse Functions - OpenStax Precalculus
  2. Testing Inverses - To verify two functions are true inverses, you plug one into the other and hope to see x both ways: f(g(x)) = x and g(f(x)) = x. If both compositions return the original input, congratulations - you've got a perfect pair! Inverse Functions - OpenStax Precalculus
  3. One-to-One & Horizontal Line Test - A function must be one-to-one (each y-value only appears once) to have an inverse, and you can check this by sliding a horizontal line across its graph. If the line ever hits more than one point, the inverse won't exist without tweaks. Inverse Functions - OpenStax Precalculus
  4. Swapping x & y to Find Inverses - Start with y = expression, swap x and y, then solve for y again - ta‑da, there's your f❻¹(x)! For y = 3x - 4, swap to get x = 3y - 4 and isolate y to complete the process. Practice makes perfect here. Inverse Functions - CourseNotes
  5. Reflection Over y = x - Graphically, inverses are mirror images across the line y = x. If you sketch f and its inverse on the same axes, you'll see this neat "reflection" effect in action. It's a visual way to confirm you've nailed the algebra. Inverse Functions - CourseNotes
  6. Common Inverse Pairs - Commit key pairs to memory, like e^x ↔ ln(x) and sin(x) ↔ arcsin(x). These familiar duos pop up everywhere in calculus, so having them down cold will save you time and trouble later! Inverse Functions Practice Questions - GeeksforGeeks
  7. Solving with Inverse Functions - When stuck in an equation like f(f❻¹(x)) = x, apply the definition to check if your solution behaves as expected. This skill is invaluable for verifying answers and cementing your understanding. Inverse Functions Practice Questions - GeeksforGeeks
  8. Domain-Range Role Swap - Remember: the domain (all possible x-values) of f becomes the range (all possible y-values) of f❻¹, and vice versa. Always double‑check these when defining or sketching inverses. Inverse Functions - OpenStax Precalculus
  9. Domain Restriction for Invertibility - Not every function is one-to-one globally. For y = x², restrict x ≥ 0 so that its inverse, √x, stays a true function. Strategic domain limits unlock inverses for many common curves. Inverse Functions - OpenStax Precalculus
  10. Real-World Applications - Inverse functions aren't just theory - they help convert Celsius to Fahrenheit or decode secret messages in cryptography. Applying concepts to real scenarios cements your mastery and keeps the math fun! Inverse Functions Practice Questions - GeeksforGeeks
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