Solving Functions and Notation Fluency

Lesson 2 of 2: HSF.IF.A.2

In this lesson:

• Solve from formulas, graphs, and tables
• Distinguish from

Learning Objectives for This Lesson

By the end, you will be able to:

4. Solve equations of the form by finding the input(s) that produce a given output
5. Read and evaluate function values from tables and graphs
6. Distinguish from

Evaluation vs. Solving: Two Directions

Evaluation: Given input , find output .

Solving: Given output , find input where .

Most function questions are one of these two directions.

Solving : Linear Function

.

Solve :

Solve :

Context: break-even at 2 units; profit hits 12 at 5 units.

Solving : Quadratic Function

.

Solve :

Solve : no real solutions

Two solutions are possible. No real solutions: is outside the range.

Quick Check: Solve

Find all such that .

Set up the equation and solve. Is there one solution or two?

Graphical Solving: Horizontal Line Method

To solve : draw a horizontal line at , find every intersection.

• Evaluate : go to , read — vertical
• Solve : go to , read — horizontal

Table Solving: Scan the Output Column

Evaluate : find , read output. Solve : scan for 15.

— two inputs, same output.

Practice: Using All Three Representations

1. . Solve algebraically.
2. Graph of : horizontal line at intersects at and . Solve .
3. From the table, solve .

Pause and answer each before the next slide.

Answers to All Three Method Problems

2. when or
3. when

All Representations Give the Same Function

vs. : The Key Distinction

• : add and first, then evaluate the function at the sum
• : evaluate at and at separately, then add the results

These are generally not equal.

Concrete Counterexample:

Compute both and compare:

Since , we have .

is not a multiplier — you cannot distribute it over addition.

Context Makes It Clear: Revenue Function

= revenue on day .

• : total from days 1 and 2
• : revenue on day 3 alone

These are different quantities — same notation, different meaning.

: Same Output, Different Inputs

: temperature after hours.

°F and °F — same output, different inputs.

The temperature passed 70°F twice: rising at hour 2, falling at hour 10.

Quick Check: Compare and

Let .

Compute both and . Are they equal?

Show both computations before you advance.

Guided Practice: Comparing Two-Function Expressions

= profit on day . = expenses.

1. Find . Interpret.
2. Find . Interpret.
3. Are these the same? Why?

Practice: Compound Function Notation Problems

Let .

1. Compute and . Equal?
2. . Find where .
3. Can for some ? Explain.

Work through each before the next slide.

Answers to Compound Notation Practice

1. ; . Not equal.
2. for all — squaring removes the sign.
3. No: is one-to-one. Only gives .

Key Takeaways from Lesson Two

• Solve : find the input — algebra, graph, or table
• Formula, table, graph all support evaluation and solving
• — is a rule, not a multiplier
• : same output, inputs need not match
• Graph: evaluate vertically, solve horizontally

Next: Key Features of Graphs

Coming up: HSF.IF.B.4

• Identify intervals where a function is increasing, decreasing, or constant
• Find maximum and minimum values from graphs
• Describe end behavior

Everything in Lessons 1 and 2 — evaluating, solving, interpreting — supports the next standard.