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Quizzes > Mathematics

Post Test: Relationships Between Functions

Quick, free function comparison quiz with instant results.

Editorial: Review CompletedCreated By: Prathmesh PatilUpdated Aug 28, 2025
Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art depicting a trivia quiz on algebraic functions for high school students.

This post test on relationships between functions helps you check how well you compare, compose, and interpret functions from graphs, tables, and rules. For more practice, try our linear and quadratic functions quiz, take a pre calculus quiz to review key ideas, or build speed with an online math test.

If f(x) = 2x + 3 and g(x) = x - 4, what is (f ∘ g)(2)?
-1
-3
-5
-7
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For f(x) = 2x + 3 and g(x) = x - 4, (f ∘ g)(x) equals 2x - 5.
False
True
undefined
A relation fails to be a function if one x-value maps to two different y-values.
True
False
undefined
Which graph feature shows that a function is not one-to-one?
It fails the vertical line test
It passes the horizontal line test
It has a y-intercept
It fails the horizontal line test
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If h(x) = (x - 1)^2 + 4, which transformation from y = x^2 produced h?
Left 4, up 1
Right 1, up 4
Left 1, up 4
Right 4, up 1
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If f(x) = 1/(x - 3), what is the domain of f?
x != 0
x >= 3
x > 3
All real numbers except x = 3
undefined
Which function has an inverse that is also a function without restricting the domain?
f(x) = |x|
f(x) = cos x
f(x) = x^3
f(x) = x^2
undefined
Suppose f maps R to R and g maps R to [0, ∞). For which composition is the domain automatically all real numbers?
f ∘ g
g ∘ f
Neither composition
Both f ∘ g and g ∘ f
undefined
If f is invertible, then f ∘ f^{-1} = id on the codomain of f.
True
False
undefined
The graphs of y = f(x) and y = f^{-1}(x) always intersect on the line y = x.
False
True
undefined
If f(x) = 2x + b and g(x) = ax + 3 are inverses, what must a equal?
1/2
-1/2
2
-2
undefined
Let f(x) = x^2 on x >= 0 and g(x) = sqrt(x). Which statement is true?
f and g are inverses on their stated domains
Neither composition equals x
f ∘ g(x) = x^2
g ∘ f(x) = x^2
undefined
If f is one-to-one and g is not, then f ∘ g cannot be one-to-one.
True
False
undefined
Suppose f: R → R is even and strictly increasing on [0, ∞). Which is true?
f has no inverse on [0, ∞)
f is not one-to-one on R
f is one-to-one on R
f is odd
undefined
If h(x) = f(g(x)) and g maps R to [2, 5], then the domain of h is limited by the domain of f restricted to [2, 5].
True
False
undefined
Consider f(x) = |x - 3| and g(x) = x - 3. Which is true?
f = g on all real x
g(x) = |f(x)|
f(x) = g(|x|)
f(x) = |g(x)|
undefined
If f and g are inverses and f is differentiable and strictly increasing, then g is strictly increasing.
False
True
undefined
Let f be one-to-one with range (2, 7). What is the domain of f^{-1}?
All real numbers
The same as the domain of f
[2, 7]
(2, 7)
undefined
If (f ∘ g)(x) is constant, then f must be constant.
False
True
undefined
Suppose f(x) = a - bx with a, b > 0 and g(x) = (x - a)/b. Which statement is true?
f and g are not inverses for any a, b > 0
f ∘ g(x) = x + a
g is the inverse of f
g ∘ f(x) = a
undefined
0

Study Outcomes

  1. Understand operations on functions such as addition, subtraction, multiplication, and division.
  2. Analyze transformations of function graphs including shifts, reflections, and scaling.
  3. Apply composite function techniques to solve complex algebraic problems.
  4. Evaluate relationships between functions using algebraic methods.
  5. Create and interpret visual representations of function operations and transformations.

Quiz: Post Test on Function Relationships Cheat Sheet

  1. Vertical & Horizontal Shifts - Moving a function up, down, left, or right is like nudging it around on the coordinate plane. Add a constant to f(x) for vertical moves, or tweak the input x for horizontal slides - this visual trick helps you predict where your graph lands. OpenStax: Transformation of Functions
  2. Reflections Over Axes - Flipping graphs is as fun as looking in a mirror: multiply f(x) by - 1 for a flip over the x-axis, or replace x with - x for a y-axis reflection. Recognizing these flips builds symmetry skills you can flex on any graph. OpenStax: Transformation of Functions
  3. Even & Odd Functions - Even functions mirror perfectly across the y-axis (f( - x)=f(x)), while odd functions spin around the origin (f( - x)= - f(x)). Spotting these patterns makes graphing a breeze and reveals hidden symmetry. OpenStax: Transformation of Functions
  4. Vertical Stretches & Compressions - Multiply f(x) by a number bigger than 1 to stretch it upwards, or by a fraction between 0 and 1 to squash it down. This "gravity tweak" changes how steep or flat your graph looks. OpenStax: Transformation of Functions
  5. Horizontal Stretches & Compressions - Tweak the input x by a constant: values between 0 and 1 stretch your graph outward, while values above 1 squeeze it in. It's like zooming in and out on your function's width. OpenStax: Transformation of Functions
  6. Combining Multiple Transformations - Stack shifts, flips, and stretches in the right order to tame complex graphs. Mastering the sequence turns chaotic shapes into neat, predictable forms. MathBits: Combo Practice
  7. Guided Practice Examples - Walk through step-by-step problems to see transformations in action. Following guided examples cements your skills and builds confidence for solo graphing adventures. GeeksforGeeks: Practice Questions
  8. Test with Practice Questions - Challenge yourself with a variety of problems covering every twist and turn of transformations. Regular quizzing hones quick recognition and sharpens your graphing intuition. GeeksforGeeks: Practice Questions
  9. Review Combined Effects - Revisit scenarios where multiple transformations overlap to see their interplay. Breaking down each step helps you untangle even the trickiest graphing puzzles. MathBits: Combo Practice
  10. Interactive Transformation Challenges - Dive into live problems that react as you tweak functions. Hands-on practice with instant feedback makes learning transformations both effective and entertaining. MathBits: Combo Practice
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