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Quizzes > High School Quizzes > Mathematics

Composite Functions Practice Quiz

Ace composition challenges with engaging examples

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Colorful paper art promoting Function Fusion Frenzy, a dynamic high school algebra quiz.

What is the correct representation of the composite function (f ∘ g)(x)?
g(f(x))
f(x) * g(x)
f(x) + g(x)
f(g(x))
The composite function (f ∘ g)(x) is defined as f evaluated at g(x). This means that g(x) is computed first and then f is applied to that result.
If f(x) = 2x + 3 and g(x) = x², what is (f ∘ g)(x)?
2x + 3
(2x + 3)²
2x² + 3
x² + 3
By substituting g(x) = x² into f, we get f(g(x)) = 2(x²) + 3, which simplifies to 2x² + 3.
If f(x) = x - 1 and g(x) = 3x, what is the value of (g ∘ f)(2)?
1
2
6
3
First, compute f(2) which equals 2 - 1 = 1. Then, apply g to get g(1) = 3 - 1 = 3.
Which statement about composite functions is true?
Composite functions are defined by adding functions together.
Composite functions are always commutative.
Composite functions are generally non-commutative.
Composite functions involve the multiplication of functions.
In general, function composition is not commutative, meaning f(g(x)) is usually not equal to g(f(x)). This makes the first option correct.
For the composite function (f ∘ g)(x) to be defined, what must be true about g(x)?
g(x) must lie within the range of f.
The domains of f and g must be identical.
f(x) must lie within the domain of g.
g(x) must lie within the domain of f.
For f(g(x)) to be defined, the output of g(x) must belong to the domain of f. This is the necessary condition for the composite function to exist.
What is the domain of (f ∘ g)(x) if f(x) = √x and g(x) = x - 3?
x > 3
x ≤ 3
x ≥ 3
All real numbers
Since f(x) = √x requires a non-negative input, we set the expression inside the square root, x - 3, to be ≥ 0. Therefore, x must be at least 3.
For f(x) = 1/(x - 2) and g(x) = x², what is the domain of (f ∘ g)(x)?
All real numbers except x = 2
All real numbers except x = √2 and x = -√2
All real numbers
x > 2 or x < -2
The function f(x) is undefined when its input equals 2. Here, f(g(x)) becomes 1/(x² - 2), so we must exclude values of x that make x² - 2 equal to 0, which happens at x = √2 and x = -√2.
Given f(x) = 3x + 4 and g(x) = x², calculate (g ∘ f)(1).
7
49
11
13
First, compute f(1) = 3(1) + 4 = 7, and then substitute into g to obtain g(7) = 7², which equals 49.
If f(x) = x² and g(x) = √x, what is (f ∘ g)(9)?
3
81
6
9
Evaluating g(9) gives √9 = 3, and then f(3) = 3² = 9. Thus, (f ∘ g)(9) equals 9.
For f(x) = 1/(x + 1) and g(x) = 2x, what is (f ∘ g)(x)?
1/(2x + 1)
2x/(x + 1)
2/(x + 1)
1/(x + 2)
By substituting g(x) = 2x into f, we obtain f(2x) = 1/(2x + 1), which is the correct expression.
Let f(x) = x² - 4 and g(x) = x + 2. What expression does (f ∘ g)(x) simplify to?
x² + 4x
x² + 2
x² - 4x
2x² + 4
Substitute g(x) = x + 2 into f to get (x + 2)² - 4. Expanding the square yields x² + 4x + 4, and subtracting 4 gives x² + 4x.
For f(x) = 3x - 1 and g(x) = x², determine (g ∘ f)(x).
9x² - 6x + 1
3x² - 1
9x² + 1
3x - 1
Substitute f(x) = 3x - 1 into g to get (3x - 1)². Expanding this expression results in 9x² - 6x + 1.
If f(x) = √x and g(x) = x + 5, what is the expression for (f ∘ g)(x)?
x + √5
x + 5√x
√x + 5
√(x + 5)
Replacing x in f(x) with g(x) = x + 5 gives f(g(x)) = √(x + 5).
Given f(x) = |x| and g(x) = x - 3, which expression correctly represents (g ∘ f)(x)?
3 - |x|
|x| - 3
|x - 3|
|x - 3| + 3
Since f(x) outputs the absolute value of x, substituting into g results in g(f(x)) = |x| - 3.
Find (f ∘ g)(2) if f(x) = x² and g(x) = 3x + 1.
7
15
49
13
First, calculate g(2): 3(2) + 1 = 7, then f(7) = 7² = 49.
If (f ∘ g)(x) = 4x + 7, where f(x) is linear in the form ax + b and g(x) = 2x - 3, find the values of a and b.
a = 1, b = 10
a = 2, b = 7
a = 2, b = 13
a = 4, b = 7
Substitute g(x) = 2x - 3 into f(x) = ax + b to get f(g(x)) = a(2x - 3) + b = 2ax - 3a + b. Equate this to 4x + 7 and solve: 2a = 4 implies a = 2, and -3a + b = 7 implies b = 13.
Given that f is invertible and (f ∘ g)(x) = h(x), how can g(x) be expressed in terms of f❻¹ and h(x)?
h❻¹(f(x))
f(h(x))
h(f❻¹(x))
f❻¹(h(x))
Since f is invertible, applying its inverse to both sides of f(g(x)) = h(x) gives g(x) = f❻¹(h(x)).
If f(x) = 2x + 1 (with inverse f❻¹(x) = (x - 1)/2) and g(x) = x², find an expression for (f ∘ g ∘ f❻¹)(x).
(x - 1)² + 1
(x + 1)²/2 + 1
(x - 1)²/2 + 1
2(x - 1)² + 1
First compute f❻¹(x) = (x - 1)/2, then substitute into g to get g(f❻¹(x)) = ((x - 1)/2)². Finally, applying f yields f(((x - 1)/2)²) = 2*((x - 1)/2)² + 1 = (x - 1)²/2 + 1.
Determine the value of x for which (f ∘ g)(x) equals (g ∘ f)(x) if f(x) = 3x + 2 and g(x) = x - 5.
No solution
x = 5
x = 13/3
x = 0
Calculating f(g(x)) gives 3(x - 5) + 2 = 3x - 13, and g(f(x)) gives (3x + 2) - 5 = 3x - 3. Setting these equal results in 3x - 13 = 3x - 3, which is a contradiction, so there is no solution.
If f(x) = 1/x and g(x) = x + 2, solve for x such that (g ∘ f)(x) = 3.
2
1
-1
0
We have (g ∘ f)(x) = g(1/x) = (1/x) + 2. Setting this equal to 3 gives (1/x) + 2 = 3, which simplifies to 1/x = 1 and therefore x = 1.
0
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Study Outcomes

  1. Analyze and evaluate composite functions through the integration of multiple function operations.
  2. Apply function notation to compute and simplify composite expressions.
  3. Determine the domain and range of composed functions in various contexts.
  4. Interpret and graph function transformations resulting from function composition.
  5. Synthesize algebraic techniques to solve real-world problems involving composite functions.

Composite Functions Cheat Sheet

  1. Understand Composite Function Basics - Composite functions are like function mash‑ups: the output of one function becomes the input of another, creating a brand‑new function! For example, if f(x)=2x+3 and g(x)=x², then f(g(x))=2x²+3, opening doors to more complex problem‑solving adventures. Ready to see more clear examples and visualizations? Explore more onlinemathlearning.com
  2. Master the Notation - Getting comfortable with (f ∘ g)(x) versus (g ∘ f)(x) is half the battle. This notation tells you exactly which function to apply first - think of it like reading directions for assembling a piece of IKEA furniture! Jump into some guided walkthroughs to cement your understanding. Dive deeper storyofmathematics.com
  3. Practice Evaluation Techniques - Substitution is your best friend when evaluating composites. If f(x)=x+5 and g(x)=3x², then (f ∘ g)(x)=3x²+5, showing how neatly one function nests inside another. Work through a variety of examples to build speedy, error‑free skills! Try examples onlinemathlearning.com
  4. Watch Those Domain Restrictions - Remember, you can only feed inputs that both functions accept - skip values that make denominators zero or square roots of negatives! Identifying the valid domain keeps your answers mathematically sound and avoids nasty "undefined" surprises. Test yourself greenemath.com
  5. Composition Isn't Commutative - Just like mixing pancake batter in the wrong order can yield a mess, (f ∘ g)(x) ≠ (g ∘ f)(x) in general. For instance, f(x)=x+1 and g(x)=x² give different results depending on which function you apply first! Embrace these quirks to level up your reasoning skills. See the proof storyofmathematics.com
  6. Break Down Complex Expressions - Decomposing h(x)=(2x+3)² into g(x)=2x+3 and f(x)=x² helps you see the inner workings of complicated formulas. This reverse‑engineering habit turns intimidating problems into bite‑sized chunks. Get the steps kristakingmath.com
  7. Tackle Triple‑Layered Compositions - When you see f(g(h(x))), unravel from the inside out: compute h(x) first, then g( ), and finally f( ). This systematic approach prevents errors and strengthens your overall function fluency. Challenge yourself greenemath.com
  8. Leverage Online Exercises - Interactive quizzes and step‑by‑step videos make practice more engaging. Hitting a mix of easy and tough problems online boosts your confidence and spots any weak points early on. Start practicing corbettmaths.com
  9. Always Apply Inner Before Outer - It's tempting to eyeball a quick answer, but sticking to the rule "inner first, outer next" keeps your results accurate every time. This disciplined habit will serve you well in exams and real‑world math alike. Review the rule storyofmathematics.com
  10. Stay Positive and Persistent - Mastery of composite functions doesn't come overnight, but a bit of daily practice and self‑belief goes a long way. Celebrate small wins, revisit tricky topics, and keep your eyes on the prize - success is just a few compositions away! Motivational tips khanacademy.org
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