9.3 Linear Functions

A linear function assigns each allowed input exactly one output using a constant rate. Algebraically it is a linear expression in one variable, f(x) = mx + b, or a linear equation in two variables with one variable named as the input. The graph is a straight line. This testing point asks you to evaluate, interpret an input/output pair in context, connect tables, graphs, and formulas, write a rule from two pairs or from one pair and a rate, and tell when a relationship is linear.

Function notation is not a new kind of math. y = 2x + 3 and f(x) = 2x + 3 describe the same line. The symbol f(x) is a label for the output. Read f(3) as the output when the input is 3, not as f times 3. This independent OpenExamPrep section uses original examples for PSAT 8/9 study at grade 8–9 difficulty.

Evaluate f(x) = mx + b

Let f(x) = 3x - 4.

• f(3) = 3(3) - 4 = 9 - 4 = 5
• f(0) = -4 (the y-intercept)
• f(-2) = 3(-2) - 4 = -6 - 4 = -10

If f(x) = 0, then 3x - 4 = 0, so 3x = 4 and x = 4/3. That input is the x-intercept of the graph. For SPR, 4/3 is three characters and is preferred over a repeating decimal.

Second function: f(x) = -2x + 9.

• f(3) = -6 + 9 = 3
• f(-4) = 8 + 9 = 17
• Solve f(x) = 1: -2x + 9 = 1, so -2x = -8, x = 4.

Check: -2(4) + 9 = -8 + 9 = 1.

What f(3) means in a story

A plant's height in centimeters after t weeks is h(t) = 40 + 8t.

h(3) = 40 + 8(3) = 40 + 24 = 64

Interpretation: after 3 weeks the plant is 64 cm tall. The 3 is time, not height. The 8 is centimeters per week (slope). The 40 is the height at week 0 (intercept).

Pages: p(n) = 8n + 20 means 20 pages already read plus 8 pages per day. Then p(3) = 8(3) + 20 = 24 + 20 = 44 pages after 3 days. If a question asks for p(3), the number is 44, not 3 and not 8.

Bowling: shoe rental $12 plus $4 per game. f(n) = 4n + 12. Then f(5) = 20 + 12 = 32 dollars for 5 games. The slope 4 is dollars per game. The intercept 12 is the rental, not the price of a game. That same slope-versus-intercept mix-up appeared in 9.2 Linear Equations in Two Variables; function notation just names the output f(n) instead of C.

When is a relationship linear?

A two-variable relationship is linear when the rate of change is constant: equal steps in x produce equal steps in y.

Linear table:

Each time x increases by 1, y increases by 3. Slope m = 3. Using the pair (1, 10): 10 = 3(1) + b, so b = 7.

f(x) = 3x + 7

Check x = 4: 3(4) + 7 = 12 + 7 = 19.

Not linear (a doubling pattern—reject the linear rule; you do not need exponential solving here):

The y-values double. The changes are +2, then +4, then +8. Not constant. Do not write f(x) = mx + b for that table. A taxi that charges $3 plus $2 per mile is linear. A culture that doubles every hour is not.

If three rows fit y = 2x + 1 and a fourth row does not, the full table is not a linear function. One stray point is enough to reject a constant-rate model on this testing point.

Slope from two points on a graph

A line passes through (0, 2) and (4, 10).

m = (10 - 2)/(4 - 0) = 8/4 = 2

Because it hits (0, 2), b = 2.

f(x) = 2x + 2

f(3) = 6 + 2 = 8

Second graph: through (2, 1) and (6, 9).

m = (9 - 1)/(6 - 2) = 8/4 = 2

Point-slope from (2, 1):

y - 1 = 2(x - 2)

y = 2x - 4 + 1

y = 2x - 3

So f(x) = 2x - 3, and f(0) = -3. Check: 2(6) - 3 = 9.

Write the rule from two input/output pairs the same way you write a line from two points. From f(1) = 6 and f(5) = 18:

m = (18 - 6)/(5 - 1) = 12/4 = 3

f(x) - 6 = 3(x - 1)

f(x) = 3x - 3 + 6 = 3x + 3

Check: 3(5) + 3 = 18.

Given slope -4 and f(2) = 1:

f(x) = -4x + b

1 = -4(2) + b

1 = -8 + b

b = 9

f(x) = -4x + 9

Check: -8 + 9 = 1.

Function notation versus y =

Questions that say the value of f(3) want 9, not the ordered pair. Questions that say which point lies on the graph of f want (3, 9).

You may see two functions. Suppose f(x) = x + 5 and g(x) = 4x. Then 3f(2) - g(2) is:

f(2) = 7, g(2) = 8

3(7) - 8 = 21 - 8 = 13

Do not compute 3f(2) as 3(2) + 5 = 11. The 3 multiplies the output f(2), after you evaluate. Another slip is 3f(2) as 3 · 2 + 5 instead of 3(2 + 5).

Trap: f(x + 1) versus f(x) + 1

These look similar and are not equal in general. Stay at grade 8–9: substitute carefully; you do not need a full transformation unit.

Let f(x) = 2x + 3.

Add 1 to the output: f(x) + 1 = (2x + 3) + 1 = 2x + 4. Every output is 1 larger (the graph shifts up 1).

Add 1 to the input: f(x + 1) = 2(x + 1) + 3 = 2x + 2 + 3 = 2x + 5. You evaluate at a different x.

Numeric check at x = 4:

• f(4) = 8 + 3 = 11
• f(4) + 1 = 12
• f(4 + 1) = f(5) = 10 + 3 = 13

Another: f(x) = 5x - 2.

• f(3) = 15 - 2 = 13
• f(3) + 1 = 14
• f(3 + 1) = f(4) = 20 - 2 = 18

If a stem asks for f(2) + 1 and you compute f(3), you answered a different question. On PSAT 8/9, that is a parentheses-and-substitution trap, not a trigonometry topic and not a circle theorem.

Putting a rule together from a rate and a point

A membership costs $20 to join and $8 per movie. Let n be movies. f(n) = 8n + 20. f(3) = 24 + 20 = 44 dollars after 3 movies. If f(n) = 68, then 8n + 20 = 68, 8n = 48, n = 6 movies. Check: 8(6) + 20 = 48 + 20 = 68.

Desmos can plot y = f(x) after you write the rule. It will not tell you whether the story was linear. Constant differences in a table will. Use the graph to confirm f(3) = 5 for f(x) = 3x - 4 by clicking x = 3, not to skip substitution practice.

/practice/psat-89Practice questions with detailed explanations