Chapter 1 Functions And Their Graphs 1.2 Exercises Answers Precalculus

You're staring at problem 47. Again. " Your highlighter is dry. The graph looks like a rollercoaster someone designed after three espressos, and the question asks: "Find the intervals on which f is increasing.Your patience is thinner.

Been there. Domain and range from a graph. 2 in most precalculus texts — Sullivan, Larson, Blitzer, OpenStax — covers graphs of functions. Piecewise-defined functions. Intercepts. Increasing, decreasing, constant intervals. Plus, section 1. Symmetry. Still, the vertical line test. Even and odd functions Simple, but easy to overlook..

It's the section where "I understand the lecture" meets "I have no idea how to start this homework."

This guide isn't an answer key. What follows is a breakdown of the core problem types in 1.Those are useless anyway — they don't teach you why the answer is what it is. 2, how to think through them, and the traps that catch almost everyone.

What Section 1.2 Actually Covers

Most precalculus textbooks align closely here. Section 1.2 is Graphs of Functions.

• Identify the domain and range of a function from its graph
• Find intercepts from a graph
• Determine intervals where a function is increasing, decreasing, or constant
• Recognize graphs of basic functions (toolkit functions)
• Graph piecewise-defined functions
• Use the vertical line test to identify functions
• Identify even and odd functions from graphs and equations

That's a lot. ** Not a picture — a map. **Reading a graph like a map.But the thread connecting all of it? Every feature tells you something about the relationship between x and y.

Domain and Range from a Graph: The Projection Trick

The concept

Domain = all possible x-values. Range = all possible y-values. From a graph, you find them by projecting the curve onto the axes.

How to do it without guessing

1. Domain: Imagine shining a light from above. The shadow on the x-axis is your domain. Look left to right. Where does the graph exist?
2. Range: Shine a light from the side. The shadow on the y-axis is your range. Look bottom to top.

Interval notation — the part where points get lost

• Open circle or asymptote? Parenthesis ( )
• Closed circle? Bracket [ ]
• Goes forever? Infinity always gets a parenthesis. Always. (-∞, 3) not [-∞, 3)

Common mistake

Writing the domain as {x | x ≠ 2} when the graph has a hole at x = 2 but continues on both sides. That's set-builder notation. Fine. But if the instructions say "use interval notation," you write (-∞, 2) ∪ (2, ∞). The union symbol ∪ matters. It means "or."

Intercepts: Don't Overthink It

x-intercepts

Where the graph crosses or touches the x-axis. y = 0.
Coordinates look like (a, 0).

y-intercepts

Where the graph crosses the y-axis. x = 0.
Coordinates look like (0, b).

The trap

A function can have multiple x-intercepts. A polynomial of degree n can have up to n. But a function can have at most one y-intercept. Why? Vertical line test. x = 0 is a vertical line. It can hit the graph once. That's it.

If a problem asks "find the intercepts" and you only give one x-value for a cubic that clearly crosses three times — you missed two. Look again.

Increasing, Decreasing, Constant: Read Left to Right

This is the #1 place students lose points. Not because they don't know the definitions. Because they read the graph wrong.

Definitions (say them out loud)

• Increasing: As x increases, y increases. Uphill left to right.
• Decreasing: As x increases, y decreases. Downhill left to right.
• Constant: As x increases, y stays the same. Flat left to right.

The protocol

1. Put your finger on the far left of the graph.
2. Move right.
3. Every time the direction changes, mark that x-value.
4. Write intervals using x-values only. Never y-values. Never coordinates.

Example

Graph goes up from x = -4 to x = -1, flat from -1 to 2, down from 2 to 5.

Increasing: (-4, -1)
Constant: (-1, 2)
Decreasing: (2, 5)

Parentheses. That said, always parentheses for increasing/decreasing/constant intervals. And the function isn't increasing at the endpoints — it's changing direction there. Some textbooks use brackets if the endpoint is included in the domain and the function is defined there. Check your syllabus. But parentheses are safer and widely accepted.

Quick note before moving on.

Piecewise Functions: Graph One Piece at a Time

The structure

f(x) = { 2x + 1    if x < 0
       { x²        if 0 ≤ x ≤ 3
       { 5         if x > 3

How to graph it without a meltdown

1. Draw light vertical lines at the boundaries (x = 0, x = 3). These are your fences.
2. Graph each piece only in its domain.
   • For 2x + 1, graph the line but only for x < 0. Stop at x = 0. Open circle at (0, 1).
   • For x², graph the parabola only from 0 to 3. Closed circles at (0, 0) and (3, 9).
   • For 5, draw a horizontal line for x > 3. Open circle at (3, 5).
3. Erase the fences.

The two questions that always appear

• "Graph the function" — do the above.
• "Evaluate f(-1), f(0), f(2), f(4)" — pick the right piece, plug in.
  • f(-1): use 2x + 1 → -1
  • f(0): use x² → 0 (note: 0 falls in second piece because of ≤)
  • f(2): use x² → 4
  • f(4): use 5 → 5

Common mistake

Using the wrong piece at the boundary. The condition with the equals sign wins. If one piece says x < 0 and the next says x ≥ 0, the second piece owns x = 0. Always Took long enough..

The Vertical Line Test: It's Not Just a Definition

You know the rule: if a vertical line hits the graph more than once, it's not a function.

What they actually test

• **Graphs that loop back

on themselves** — a sideways parabola or a circle will fail immediately, and the question often hides it inside a "which of these is a function?Practically speaking, " multiple choice where three options look innocent. Here's the thing — no vertical line needed; the repetition is the tell. That's why - Tables with repeated x-values and different y-values — same logic applies. That said, if x = 2 maps to both 5 and 7, it's not a function. - Set-builder or mapping notation — check whether any input has two outputs. If it does, the relation fails the test even without a picture Took long enough..

Quick drill

Look at the graph. Imagine dropping a ruler straight down anywhere along the x-axis. If it ever touches the curve in two places at once, write "not a function" and move on. Don't overthink the shape — the test is binary.

Domain and Range: Read the Axes, Not Your Assumptions

Students default to "all real numbers" because it's fast. Sometimes it's right. Often it isn't That's the part that actually makes a difference..

From a graph

• Domain = all x-values the graph covers, left to right.
• Range = all y-values the graph covers, bottom to top.
• Use brackets [ ] when the endpoint is included (solid dot, or the graph runs off the grid with arrows). Use parentheses ( ) for holes (open circles) or asymptotes the graph never reaches.

From an equation

• Fraction: denominator ≠ 0. Set it equal to zero, solve, exclude.
• Square root: radicand ≥ 0. Set inside ≥ 0, solve.
• Both: intersect the restrictions.

Example: f(x) = √(x - 3) / (x - 5)
Domain: x ≥ 3 and x ≠ 5 → [3, 5) ∪ (5, ∞)

Conclusion

Function analysis is less about advanced math and more about disciplined reading. Left to right for behavior. Boundaries first for piecewise. Vertical lines for function status. Axes for domain and range. The errors are predictable, which means they're preventable — slow down at the endpoints, check the condition with the equals sign, and never trust a default answer. Do that consistently and the points stop disappearing Easy to understand, harder to ignore..