Precalculus Study Guide: Inverse and Composite Functions, One-to-One Functions, and Graph Interpretation

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Q1. Use the graphs of f and g to find (f ∘ g)(x - 3).

Background

Topic: Composite Functions and Graph Interpretation

This question tests your ability to evaluate composite functions using graphs. Specifically, you are asked to find the value of (f ∘ g)(x - 3), which means you must first evaluate g at (x - 3), then use that result as the input for f.

Key Terms and Formulas

Composite Function:
To find , first compute , then plug that value into .

Graphs of f(x) and g(x)

Step-by-Step Guidance

Identify the value of you want to use. If a specific $x$ is not given, choose a value or leave it as a variable for now.
Find using the graph of . Locate the point on the $g(x)$ graph where the input is and read the corresponding output value.
Take the output from and use it as the input for . That is, find by locating this value on the graph.
Set up the final expression for using the values you found, but do not compute the final answer yet.

Try solving on your own before revealing the answer!

Final Answer: (f ∘ g)(x - 3) = 1

For the value of shown in the graph, yields a specific value, and plugging that into gives the final result of 1.

Q2. Use the graphs of f and g shown below to evaluate (f ∘ g)(1).

Background

Topic: Composite Functions and Graph Reading

This question asks you to evaluate the composite function using the provided graphs of and .

Key Terms and Formulas

Composite Function:
First, find , then use that value as the input for .

Graphs of f(x) and g(x) for composite evaluation

Step-by-Step Guidance

Locate on the graph of and determine the value of .
Take the value you found for and use it as the input for . That is, find by locating this value on the $f(x)$ graph.
Set up the expression for using the values you found, but do not compute the final answer yet.

Try solving on your own before revealing the answer!

Final Answer: (f ∘ g)(1) = 2

By reading the graphs, gives a certain value, and evaluated at that value gives the final answer of 2.

Q3. Choose the graph that shows a one-to-one function and its inverse.

Background

Topic: One-to-One Functions and Inverses

This question tests your understanding of one-to-one functions and their inverses, as well as your ability to recognize them graphically. A function is one-to-one if every -value corresponds to exactly one -value. The graph of a function and its inverse are reflections across the line .

Key Terms and Formulas

One-to-One Function: Passes the horizontal line test (no horizontal line intersects the graph more than once).
Inverse Function: If is a function, its inverse satisfies and .
Graphical Property: The graph of is the reflection of across the line .

Graphs showing functions and their inverses

Step-by-Step Guidance

Review each graph and identify which ones show a function and its inverse. Look for a blue solid curve (the function) and a red dashed curve (the inverse).
Check if the blue curve passes the horizontal line test (no horizontal line crosses it more than once).
Verify that the red dashed curve is a reflection of the blue curve across the line .
Identify the graph that meets both criteria: the function is one-to-one, and the red dashed curve is its inverse.

Try solving on your own before revealing the answer!

Final Answer: Graph A

Graph A shows a blue solid curve (the function) that passes the horizontal line test and a red dashed curve (the inverse) that is a reflection across .