BetterGrades Precalculus · Unit 2 · Lesson

Function notation, evaluation, and solving

Use function notation accurately and distinguish finding an output from finding inputs that produce a chosen output.

Opening

Start with the situation

The notation $f(x)$ names the output of function $f$ at input $x$ ; it is not multiplication.

The same dependency may arrive as a story, table, graph, or formula. Learning to preserve the inputs, outputs, units, and domain while moving among those forms is the central language of the course.

Before you begin

Prerequisite check

Use function notation from the algebra and function readiness unit.
Read ordered pairs and interval notation.
Attach units to contextual quantities.

Core explanation

Explanation

Substitute the entire input with parentheses or trace the input through a table or graph. For $f(x)=k,$ solve for every preimage.

A function produces at most one output for an allowed input, but a fixed output may have several inputs.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through vertical versus horizontal tracing, substitution grouping, or another equivalent representation.

Conceptual reading

What the idea is really doing

A function is a dependency, not merely an equation. Inputs, outputs, units, and domain must agree in words, tables, graphs, and formulas; each representation should tell the same mathematical story.

This lesson narrows that lens to one goal: use function notation accurately and distinguish finding an output from finding inputs that produce a chosen output. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Substitute the entire input with parentheses or trace the input through a table or graph. For $f(x)=k$ .
Solve for every preimage.
Where is the same feature visible in another representation?.

Verification: Choose two representations and make them verify one another. A table can test a formula, a graph can expose a domain or range claim, and units can reveal a model that is algebraically neat but conceptually wrong.

Foundation walkthrough

Plan before calculating

Problem

For $f(x)=2x^2-3x,$ find $f(-2)$ .

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Substitute the entire input with parentheses or trace the input through a table or graph. For $f(x)=k,$ solve for every preimage.
Conclusion: $14$
Why the check works: Parentheses preserve the negative input.

Worked examples

See the idea in three forms

foundation example

For $f(x)=2x^2-3x,$ find $f(-2)$ .

Solution $14$

Parentheses preserve the negative input.

representation example

For $g(x)=x^2+1,$ find $g(-3)$ .

Solution $10$

This example expresses function notation, evaluation, and solving in a second form.

transfer example

For $f(x)=x^2,$ solve $f(x)=16$ .

Solution $x=\pm 4$

A function produces at most one output for an allowed input, but a fixed output may have several inputs.

Vertical versus horizontal tracing. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Parentheses preserve the negative input. — **Anchor figure · Vertical versus horizontal tracing**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Parentheses preserve the negative input.

Substitution grouping. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for function notation, evaluation, and solving. — **Mechanism figure · Substitution grouping**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for function notation, evaluation, and solving.

Preimage set. Compare the valid path with the tempting shortcut. The figure shows why confusing f(a) with the equation f(x)=a leads to a false conclusion. — **Comparison and error figure · Preimage set**
Compare the valid path with the tempting shortcut. The figure shows why confusing $f(a)$ with the equation $f(x)=a$ leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is confusing $f(a)$ with the equation $f(x)=a$ .

Check yourself

For $f(x)=4x-7,$ find $f(5)$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

For $f(x)=4x-7,$ find $f(5)$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 202

For $g(x)=x^2+1,$ find $g(-3)$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 303

For $f(x)=x^2,$ solve $f(x)=16$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 404

How solve $f(x)=0$ from a graph?

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 505

Explain why this conclusion is valid: $14$ . Use the foundation problem as evidence: For $f(x)=2x^2-3x,$ find $f(-2)$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 606

Solve the representation example, then name the feature of function notation, evaluation, and solving that it illustrates: For $g(x)=x^2+1,$ find $g(-3)$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is confusing $f(a)$ with the equation $f(x)=a$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 808

Connect two representations for this example: For $f(x)=2x^2-3x,$ find $f(-2)$ . Describe what a graph, table, mapping, or algebraic form would have to show.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 909

Create a nearby example by changing one number or condition in this prompt: For $f(x)=x^2,$ solve $f(x)=16$ . Predict the effect, solve your new example, and compare it with the original.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 1010

Write a short verification checklist for function notation, evaluation, and solving, then apply it to one worked example from this lesson.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Lesson close

Connect forward

The next lesson, Words, tables, graphs, and formulas, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Yoshiwara, Modeling, Functions, and Graphs
Lippman and Rasmussen, Precalculus Volume 1
Stitz and Zeager, Precalculus

No long source passage is reproduced.

For f(x)=2x2−3x,f(x)=2x^2-3x,f(x)=2x2−3x, find f(−2)f(-2)f(−2).

For f(x)=2x2−3x,f(x)=2x^2-3x,f(x)=2x2−3x, find f(−2)f(-2)f(−2).

For g(x)=x2+1,g(x)=x^2+1,g(x)=x2+1, find g(−3)g(-3)g(−3).

For f(x)=x2,f(x)=x^2,f(x)=x2, solve f(x)=16f(x)=16f(x)=16.

Original BetterGrades manuscript, rights-separated references.

For $f(x)=2x^2-3x,$ find $f(-2)$ .

For $f(x)=2x^2-3x,$ find $f(-2)$ .

For $g(x)=x^2+1,$ find $g(-3)$ .

For $f(x)=x^2,$ solve $f(x)=16$ .