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Quizzes > High School Quizzes > Electives

Relationships Between Functions Practice Quiz

Master function relationships with our practice test

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art depicting a trivia quiz on algebraic functions for high school students.

If f(x) = x + 2, what is f(3)?
3
1
5
6
To find f(3), substitute 3 in place of x, giving 3 + 2 = 5. This demonstrates basic function evaluation.
If g(x) = 2x, what is g(4)?
12
10
8
6
By substituting x = 4 into g(x), we get 2 Ã - 4 = 8. This reinforces the concept of direct substitution in function evaluation.
What transformation is described by f(x) + 3?
Vertical stretch
Horizontal shift right by 3
Vertical shift upward by 3
Reflection across the x-axis
Adding 3 to f(x) shifts the graph vertically upward by 3 units. This is a basic vertical translation of the function.
What is the domain of the function f(x) = √(x - 1)?
x ≥ 1
x > 1
All real numbers
x ≤ 1
For √(x - 1) to be defined, the expression under the square root must be non-negative, hence x - 1 ≥ 0. This gives the domain x ≥ 1.
Which expression represents the composition of two functions f and g?
g(f(x))
f(x) + g(x)
f(g(x))
f(x) * g(x)
The composition f(g(x)) means that the output of g(x) is used as the input for f(x). This is the standard definition of function composition.
Given f(x) = x² and g(x) = x + 1, what is (f + g)(2)?
9
7
8
6
Calculate f(2) as 2² = 4 and g(2) as 2 + 1 = 3. Adding these gives 4 + 3 = 7.
If f(x) = 3x and g(x) = x - 4, what is (f ∘ g)(x)?
3x - 12
x + 4
x - 12
3x + 4
The composition f(g(x)) involves substituting g(x) into f, giving f(x - 4) = 3(x - 4) = 3x - 12. This follows the definition of composition.
What is the effect on the graph of f(x) when it is replaced by f(x - 2)?
Shift right by 2
Shift left by 2
Shift up by 2
Vertical stretch
Replacing x with (x - 2) shifts the graph horizontally to the right by 2 units. This follows the rule for horizontal translations.
If f(x) = |x|, what is the value of f(-5)?
10
5
-5
0
The absolute value of -5 is 5. This question tests the basic property of the absolute value function.
What is the inverse of the function f(x) = x + 7?
7 - x
x - 7
1/(x + 7)
x + 7
To find the inverse, swap x and y in the equation y = x + 7 and solve for y, giving y = x - 7. The inverse function undoes the original function.
Given f(x) = 2x² - 3, what is f(0)?
0
3
2
-3
Substitute x = 0 into the function: f(0) = 2(0)² - 3 = -3. This demonstrates evaluating a quadratic function at a specific point.
For h(x) = f(x) + g(x) where f(x) = 3x and g(x) = x², what is h(2)?
14
12
8
10
Evaluate f(2) as 3(2) = 6 and g(2) as (2)² = 4. Therefore, h(2) = 6 + 4 = 10.
Which expression represents a vertical compression of f(x) by a factor of 1/2?
f(1/2 · x)
1/2 · f(x)
2 · f(x)
f(x) - 1/2
Multiplying the function f(x) by 1/2 compresses the graph vertically by a factor of 1/2. This transformation scales down all y-values.
If f(x) = x² and it is reflected across the x-axis, what is the new function?
-x² - 1
x²
x² + 1
-x²
Reflecting a function across the x-axis changes the sign of the output, so f(x) = x² becomes -x². Reflections invert the graph over the x-axis.
What does the composition f(g(x)) represent in terms of function operations?
f(x) is applied to x first, then g is applied to the result
g(x) is applied to x first, then f is applied to the result
The product of f(x) and g(x)
The sum of f(x) and g(x)
The composition f(g(x)) means that g(x) is computed first and its output is then used as the input for f(x). This is a fundamental property of function composition.
Let f(x) = 2x + 3 and g(x) = x². What is f(g(-2))?
7
-5
11
1
First, compute g(-2) = (-2)² = 4. Then, substitute into f to get f(4) = 2(4) + 3 = 11. This two-step process illustrates composition evaluation.
Consider f(x) = |x - 4| and g(x) = x + 1. For what value of x does f(g(x)) equal 0?
3
4
-3
1
Compute f(g(x)) = |(x + 1) - 4| = |x - 3|. The absolute value equals 0 only when x - 3 = 0, so x = 3. This problem combines composition with solving an absolute value equation.
Given h(x) = f(x) - g(x) where f(x) = 3(x - 2) and g(x) = x + 1, what is h(5)?
6
2
15
3
Evaluate f(5) = 3(5 - 2) = 9 and g(5) = 5 + 1 = 6, then compute h(5) = 9 - 6 = 3. This problem requires combining and simplifying two functions.
A function f is given by f(x) = a(x - h)² + k. If the vertex is (2, -3) and the graph has a vertical stretch by a factor of 4, which is the correct form of f(x)?
4(x - 2)² + 3
4(x + 2)² - 3
(x - 2)² - 3
4(x - 2)² - 3
The vertex form of a quadratic is f(x) = a(x - h)² + k. With vertex (2, -3) and a vertical stretch factor of 4, a equals 4, yielding f(x) = 4(x - 2)² - 3. The sign inside the parenthesis reflects the vertex's x-coordinate.
Consider f(x) = x² and g(x) = 2x. What is (f ∘ g)(x) and what transformation does it represent compared to f(x)?
2x², which is a horizontal stretch by a factor of 2
2x², which is a vertical stretch by a factor of 2
4x², which is a horizontal stretch by a factor of 1/2
4x², which is a vertical stretch by a factor of 4
Calculating f(g(x)) gives f(2x) = (2x)² = 4x². Compared to f(x) = x², the coefficient 4 implies a vertical stretch by a factor of 4. This shows how composition can alter the graph's scale.
0
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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. 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. 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. 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. 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. 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. 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. 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. 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. 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
    11. MathBits: Combo Practice
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