Reading a Function from Its Graph: f(−1), f(2), Zeros, Domain & Range (Stewart Ch.1 Q1)

Direct answer. f(−1) = −2. f(2) ≈ 2.8. The equation f(x) = 2 has two solutions, x = −3 and x = 1. The zeros (where f(x) = 0) are approximately x ≈ −2.3 and x ≈ 0.3. The domain is [−3, 3] and the range is [−2, 3].

Why Stewart starts the entire textbook here

Stewart opens with a graph because every calculus topic that follows — limits, continuity, derivatives, integrals, the entire study of how functions behave — depends on the student's ability to read a graph without thinking about it. The student who can look at a curve and immediately tell you a function value, the inputs that produce a given output, the zeros, and the boundaries of the domain has the visual fluency that calculus assumes. The student who cannot do this slows down on every later problem because each one starts by interpreting a picture.

Part (a) tests function notation. Part (b) tests estimation. Part (c) tests solving an equation by intersection. Part (d) tests finding zeros. Part (e) tests domain and range as intervals. Five questions, five different reflexes, all from the same picture.

(a) State the value of f(−1)

Function notation: f(−1) means "the output of f when the input is −1." On a graph that is the y-coordinate of the point on the curve directly above (or below) x = −1.

Trace a vertical line up from x = −1 on the x-axis until you hit the curve. Then read horizontally to the y-axis. The point on the curve sits at y = −2.

f(−1) = −2

This is exact, not an estimate. The point lands on a grid intersection.

(b) Estimate the value of f(2)

Same move — trace up from x = 2 until you hit the curve, then read across. This time the curve does not pass through a grid intersection, so we have to estimate. The point sits a little above y = 2.5 but well below y = 3 — about four-fifths of the way from 2 to 3.

f(2) ≈ 2.8

Stewart cues the word "estimate" deliberately. The right answer is "about 2.8" — confident to the tenths place, not to the hundredths. Students who write f(2) = 2.83 are over-claiming what a graph can give you. Students who write f(2) ≈ 3 are under-using the gridlines.

(c) For what values of x is f(x) = 2?

This is the reverse direction of part (a). Now we are given the output and asked for the inputs. Geometrically: where does the curve reach the height y = 2?

Draw (or imagine) a horizontal line at y = 2 running across the whole graph. Every x-coordinate where that horizontal line touches the curve is a solution. The line at y = 2 intersects the curve in two places — at x = −3 (the leftmost endpoint, which sits at y = 2) and at x = 1 (where the curve climbs back up through height 2 on the right side).

x = −3 and x = 1

This is the visual model behind solving any equation of the form f(x) = k: draw the horizontal line y = k, count intersections, read off x-values. Later in the textbook this is exactly how Stewart introduces the Intermediate Value Theorem.

(d) Estimate the values of x such that f(x) = 0

The zeros of a function are the x-values where the curve crosses the x-axis. This is part (c) with k = 0 — the horizontal line we're chasing is the x-axis itself.

The curve crosses the x-axis twice. Once between x = −3 and x = −2, closer to −2 — about −2.3. Once between x = 0 and x = 1, closer to 0 — about 0.3.

x ≈ −2.3 and x ≈ 0.3

Again the word "estimate" is the signal — read to the tenths place, not finer. The reason this skill matters is that later in Stewart, zeros become the foundation of factoring, root-finding (Newton's method), and analyzing the sign of derivatives.

(e) State the domain and range

The domain is the set of x-values the function actually takes — the horizontal extent of the graph. The range is the set of y-values the function actually outputs — the vertical extent.

Domain. The graph starts at x = −3 on the left and ends at x = 3 on the right. Both endpoints are closed (solid dots), so they are included.

Domain: [−3, 3]

Range. The lowest y-value the curve reaches is −2 (at the bottom of the dip). The highest y-value is 3 (at the right endpoint). Both are attained, so both endpoints are included.

Range: [−2, 3]

Square brackets mean "endpoint included." If an endpoint were drawn as an open circle, you'd use a parenthesis on that side: [−3, 3) would mean "−3 included, 3 excluded." Stewart's Q1 uses solid dots on both ends, so it's brackets on both sides.

The three mistakes worth memorizing

• Reading the wrong coordinate. f(−1) is a y-value, not an x-value. Students sometimes write f(−1) = −1 because the input is −1 — they read the input back instead of the output. The fix: always say out loud, "I'm looking up the input −1, and I want the output, which is the y-value of the curve there."
• Confusing "zeros of f" with "y-intercept." Zeros are x-values where f(x) = 0 — points where the curve crosses the x-axis. The y-intercept is the single output value at x = 0. They live on perpendicular axes and they are not the same thing.
• Swapping domain and range. Domain comes from the x-axis (horizontal extent); range comes from the y-axis (vertical extent). A useful memory hook: "domain" is the input you give, like the domain of a website is what you type in. Range is what comes out. Inputs are usually x, outputs are usually y.

How SOMATH teaches this problem

We teach Stewart Ch.1 Q1 as a five-station drill. Same picture, five different reflexes: read forward (input → output), read backward (output → input), find zeros, write the domain, write the range. Students cycle through all five operations on the same graph until each one is automatic, then we change the graph and run it again. By the end of the lesson students can look at any function plot and answer all five questions in under thirty seconds — which is exactly the speed Stewart's later chapters assume.

Our classroom at 226 W 79th St (Upper West Side, between Broadway and Amsterdam) runs small groups of three to five students with whiteboards on every wall. We don't hand out worksheet stacks. Each technique gets introduced with the why, drilled on three increasingly hostile graphs, and tied back to the calculus context where the student will meet it next. This problem is the gate every AP Calc and pre-calc student walks through at SOMATH.

FAQ

What is the difference between f(−1) = −2 and f(x) = 2?

The first asks "what is the output when the input is −1?" — you give it an x, it gives you back a y. The second asks "what inputs produce the output 2?" — you give it a y, it gives you back all the x's that map there. Forward versus backward, single output versus possibly many inputs.

How accurate does an estimate from a graph need to be?

Read to the tenths place — one digit past whatever the gridlines show. Anything more precise is over-claiming. For Stewart Ch.1 Q1, the gridlines are at integer values, so estimates to one decimal place (2.8, −2.3, 0.3) are what's expected.

What does the closed dot at the endpoint of the graph mean?

A closed (solid) dot at an endpoint means that point is included in the function. An open circle would mean it's excluded. For Stewart Ch.1 Q1, both endpoints are closed, so the domain and range are both written with square brackets.

Will this kind of problem show up on the AP Calculus exam?

Yes, constantly — though usually as the setup for something else. AP Calc free-response problems frequently begin with a graph and ask you to read values, identify intervals where the function is increasing, or estimate a derivative or area from the picture. The skill Stewart builds in Ch.1 Q1 is the foundation of every "given the graph of f…" prompt on the exam.

What is the next Stewart problem after this one?

Stewart Ch.1 Q2 hands you the graphs of two functions and asks you to read values from each, then combine them. Same set of reflexes, applied to two curves at once. We have walkthroughs of those linked at the bottom of this post.

Related posts

• Stewart Q4: Two Points — Line, Midpoint, and Circle with AB as Diameter
• Stewart Q2: Equation of a Circle from Center and a Point
• Stewart Q1 (Analytic Geometry): Equation of a Line from Slope and a Point
• Functions and Their Domains in AP Pre-Calculus
• Find the Domain: Rational, Cube & Square Root (Ch.1 Q3)