Algebra 1
Introducing Functions
Fresh, parallel-form problems with full worked solutions — more reps for the skills in this unit, kept separate from the textbook's own problems.
Lesson 4.1: What is a function
Is {(3,9),(6,9),(9,9)} a function? Say yes or no, and name the reason.
Worked solutionTry it first, then open.
Look only at the inputs: 3, 6, 9 — each appears once, so each input is paired with exactly one output. The output 9 repeating is fine (that's the "different buttons, same snack" case — repeats are only a problem on the input side). Yes, it is a function.
Answer: Function — outputs repeat (all 9) but the inputs 3, 6, 9 are distinct, so each input has exactly one output.
Is {(7,2),(8,4),(7,5)} a function? Say yes or no, and name the input that breaks it.
Worked solutionTry it first, then open.
Scan the inputs: 7 shows up twice — once paired with 2 and once with 5. That's one input with two different outputs, which is exactly what a function may not do. The input 7 breaks it, so this is not a function. (Contrast: a repeated output would have been fine; it's the repeated input with conflicting outputs that fails.)
Answer: Not a function — input 7 is paired with both 2 and 5 (one input, two outputs).
A table has inputs 5, 6, 5, 7 and outputs 0, 1, 8, 1. Is it a function? Explain.
Worked solutionTry it first, then open.
Line the pairs up: (5,0), (6,1), (5,8), (7,1). The input 5 appears twice and gives two different outputs, 0 and 8 — one input, two outputs. So it is not a function. The repeated output 1 (from inputs 6 and 7) is allowed; only the split input 5 matters.
Answer: Not a function — input 5 appears twice with outputs 0 and 8.
A graph is the horizontal line y=4 (flat, going left to right). Is it a function? Use the vertical line test to explain — and say what its one output always is.
Worked solutionTry it first, then open.
Sweep an imaginary vertical line across the graph. At every position it crosses the flat line y=4 in exactly one point, so each input x has exactly one output. The vertical line test passes, so it is a function. Its output is always 4, no matter the input — this is a constant function. (Don't confuse it with the vertical line x=4, which is NOT a function: there a single input has infinitely many outputs.)
Answer: Yes, it is a function. The vertical line test passes: every vertical line hits a horizontal line in exactly one point, so each input has exactly one output.
Lesson 4.2: Function notation f(x), domain & range
With f(x)=4x−3, find f(3).
Worked solutionTry it first, then open.
Substitute 3 for x, keeping it in parentheses: f(3)=4(3)−3. Multiply first: 4(3)=12. Then subtract: 12−3=9. So f(3)=9. (Remember f(3) means "run 3 through the rule f," not f times 3.)
Answer: 9
With g(x)=x²−2, find g(−3).
Worked solutionTry it first, then open.
Substitute with parentheses around the input: g(−3)=(−3)²−2. Square the whole input: (−3)²=9 (a negative times a negative is positive — not −9). Then 9−2=7. So g(−3)=7.
Answer: 7
With q(x)=2−5x, find q(−2).
Worked solutionTry it first, then open.
Substitute −2 in parentheses: q(−2)=2−5(−2). Multiply first: −5(−2)=+10 (negative times negative is positive). Then 2+10=12. So q(−2)=12. The parentheses are what keep the sign honest.
Answer: 12
With f(x)=4x−3, evaluate at the expression 2x: find f(2x). (Light stretch — feed an expression into the rule.)
Worked solutionTry it first, then open.
Whatever sits in the parentheses goes everywhere x was. So f(2x)=4(2x)−3. Multiply the constants: 4·2x=8x. That leaves f(2x)=8x−3. Nothing else combines, so the answer is 8x−3.
Answer: 8*x-3
Lesson 4.3: Multiple representations; linear vs. nonlinear
Build the table for f(x)=6x at x=0,1,2,3,4. What is the last output, f(4)? (Also: is the table linear?)
Worked solutionTry it first, then open.
Plug in each input: f(0)=0, f(1)=6, f(2)=12, f(3)=18, f(4)=6(4)=24. So the last output is f(4)=24. The outputs step up by a constant +6 each time (6−0, 12−6, 18−12, 24−18 are all 6), so the table is linear with rate of change +6.
Answer: 24
For the table x: 0,1,2,3 → y: 7,4,1,−2, the inputs step evenly. Is it linear or nonlinear, and what is its rate of change?
Worked solutionTry it first, then open.
First confirm the inputs step evenly: 0,1,2,3 go up by +1 each time. Now check the output differences: 4−7=−3, 1−4=−3, −2−1=−3 — all the same. Equal x-steps give equal output steps, so it is linear, with a constant rate of change of −3. (As a preview of Unit 5, that constant −3 is the slope, and the value 7 at x=0 gives the equation y=−3x+7.)
Answer: Linear. Equal x-steps of +1 give equal output steps of −3 (4−7, 1−4, −2−1 are all −3), so the rate of change is constant at −3.
From g(x)=4x−5, build the table at x=0,1,2,3. What is g(3)?
Worked solutionTry it first, then open.
Run each input through the rule: g(0)=4(0)−5=−5, g(1)=4(1)−5=−1, g(2)=4(2)−5=3, g(3)=4(3)−5=12−5=7. So g(3)=7. The outputs −5,−1,3,7 climb by a constant +4 each step, matching the 4 in front of x — linear.
Answer: 7
A table's outputs go 1, 6, 13, 22 for x = 1, 2, 3, 4. Linear or nonlinear? Tell how you know without graphing. (Light stretch.)
Worked solutionTry it first, then open.
The inputs 1,2,3,4 step evenly by +1, so it's fair to compare output differences. Those differences are 6−1=5, 13−6=7, 22−13=9 — the steps 5,7,9 are not equal, they keep growing. Since equal x-steps do not give equal output steps, the rate of change is not constant, so the function is nonlinear; its graph would curve upward rather than be a straight line. (It is still a function — nonlinear just means "not a straight line," not "broken.")
Answer: Nonlinear. Output steps are 5, 7, 9 (not constant), so the rate of change changes — the graph curves rather than forming a straight line.
Mixed review
Problems that mix skills from across the unit — good for spacing earlier work back in.
With h(x)=2x+5, find h(−4).
Worked solutionTry it first, then open.
Substitute with parentheses: h(−4)=2(−4)+5. Multiply first: 2(−4)=−8. Then −8+5=−3. So h(−4)=−3. (This is lesson 4.2's evaluate-at-a-negative skill — and a free negatives + order-of-operations callback to Units 1–2.)
Answer: -3
Is {(0,5),(1,5),(2,8),(0,9)} a function? Say yes or no, name the input that breaks it, and say why the repeated output is not a problem.
Worked solutionTry it first, then open.
Check the inputs: 0 appears twice, paired with 5 and with 9 — one input giving two outputs, which a function may not do, so it is not a function and input 0 is the culprit. The output 5 showing up for both 0 and 1 is allowed, because repeats are only a problem on the input side (two buttons can give the same snack). This interleaves lesson 4.1's one-output-per-input rule with the repeated-output trap.
Answer: Not a function — input 0 is paired with both 5 and 9 (one input, two outputs). The output 5 repeating across inputs 0 and 1 is fine; the split input 0 is what breaks it.
With g(x)=x²+x, find g(−3).
Worked solutionTry it first, then open.
Substitute −3 in parentheses everywhere x appears: g(−3)=(−3)²+(−3). Square the whole input: (−3)²=9. Then add the second term: 9+(−3)=6. So g(−3)=6. Watch the squared term — (−3)²=9, not −9. (Lesson 4.2 sign care, with a quadratic rule.)
Answer: 6
A table has x: 0,1,2,3 → y: 1,3,7,15. The inputs step evenly. Linear or nonlinear, and how can you tell?
Worked solutionTry it first, then open.
Inputs 0,1,2,3 step by +1 each time, so compare output differences: 3−1=2, 7−3=4, 15−7=8. The steps 2,4,8 are not equal (they double), so the rate of change is not constant and the function is nonlinear — its graph curves. (This doubling previews exponential growth in Unit 9.) This mixes lesson 4.3's linear-vs-nonlinear test with the "check x-steps are even first" habit.
Answer: Nonlinear. With even x-steps of +1, the output steps are 2, 4, 8 (a doubling pattern), which are not constant — so the rate of change is not constant and the graph is not a straight line.
The linear table x: 0,1,2,3 → y: −1,3,7,11 has even x-steps. Read off its equation, then use the equation to confirm the output at x=3. (Light stretch — table → equation → check.)
Worked solutionTry it first, then open.
Find the constant step: 3−(−1)=4, 7−3=4, 11−7=4 → +4 per step, so 4 sits in front of x. The value at x=0 is −1, the constant added. That gives the equation y=4x−1. Now confirm against the last row using the equation: at x=3, 4(3)−1=12−1=11, which matches the table's y=11. This interleaves lesson 4.3's table→equation move with evaluating to check (lesson 4.2).
Answer: 11