GED Math: Patterns and Functions
The domain of a function is the set of all input values that make sense. The range is the set of all output values the function can produce. To find the domain, look for values that must be excluded: division by zero is undefined, and a square root of a negative number is not a real number. Real-world situations can limit the domain too (time cannot be negative). To find the range, think about the smallest and largest outputs: a squared expression is never negative, an absolute value is never negative, and a negative sign in front flips the result. If you are unsure, graph the function or try several inputs and look at how the outputs behave.
1. Patterns and Sequences
A pattern (or sequence) is a list of numbers that follows a rule. Each number is a term. To find the rule, compare each term with the one before it. If the difference is always the same, the pattern adds (or subtracts) a fixed number each time. If the terms are multiplied or divided by the same number each time (a constant ratio), the pattern is multiplicative. Some patterns alternate signs. Once you know the rule, apply it again and again to reach later terms. Always test the rule on every pair of terms before you trust it. Test tip: ask yourself, what operation (or sequence of operations) turns each term into the next one?
Worked Example: Find the seventh term of 3, 6, 12, 24, 48, …
Each term is 2 times the one before it. The sixth term is 48 × 2 = 96, and the seventh is 96 × 2 = 192.
1. What number comes next in the pattern −12, −9, −6, −3, ___?
Explanation: The terms increase by 3 each time, so −3 + 3 = 0.
2. What is the next number in the sequence 21, 26, 31, 36, ___?
Explanation: The terms go up by 5, so 36 + 5 = 41.
3. What is the next term in 1000, 500, 250, 125, 62.5, ___?
Explanation: Each term is half the one before, so 62.5 ÷ 2 = 31.25.
4. What is the next number in 3, −5, 7, −9, 11, ___?
Explanation: The absolute values go up by 2 and the signs alternate: after +11 comes −13.
5. What is the seventh term of 3, 6, 12, 24, 48, …?
Explanation: Double each time: 96 is the sixth term, 192 is the seventh.
6. What is the sixth term of −14, −8, −2, 4, …?
Explanation: The terms go up by 6: 4, 10, 16. The fifth term is 10, and the sixth is 16.
7. What is the next term in 1, 4, 9, 16, 25, ___?
Explanation: These are the perfect squares 12, 22, 32, … The next is 62 = 36.
8. The 1st, 2nd, 3rd, and 4th figures use 1, 3, 5, and 7 blocks. How many blocks does the 25th figure need?
Explanation: The number of blocks is 2n − 1, so for n = 25 it is 2(25) − 1 = 49.
2. What Is a Function? Inputs, Outputs, and Evaluating
A function is a rule that matches each input (often x) with exactly one output (often y). The output depends on the input. A function can be shown as a table, a graph, a sentence, or an equation. For example, if a vendor charges $1.75 per slice of pizza, the cost C depends on the number of slices n: C = 1.75n. To evaluate a function, substitute the given input for the variable (use parentheses, especially with negatives), then follow the order of operations. Remember that every input has one output, so a table that gives the same input two different outputs is not a function.
Worked Example: In the function y = 4x − 7, find y when x = −2.
Substitute: y = 4(−2) − 7 = −8 − 7 = −15.
| Slices | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Cost | $1.75 | $3.50 | $5.25 | $7.00 | $8.75 | $10.50 |
1. A vendor charges $1.75 per slice (see the table). Which function gives the cost C of n slices?
| Slices | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Cost | $1.75 | $3.50 | $5.25 | $7.00 | $8.75 | $10.50 |
Explanation: Cost is the price per slice times the number of slices: C = 1.75n.
2. Using C = 1.75n, what is the cost of 8 slices?
Explanation: C = 1.75(8) = 14.00.
3. In the function y = 3x + 5, what is y when x = −4?
Explanation: y = 3(−4) + 5 = −12 + 5 = −7.
4. In the function y = 2x(4 + x) − 2, what is y when x = 3?
Explanation: y = 2(3)(4 + 3) − 2 = 6(7) − 2 = 40.
5. In the function y = x + 36 − 8, what is y when x = 21?
Explanation: y = (21 + 3)/6 − 8 = 24/6 − 8 = 4 − 8 = −4.
6. In the function y = (x − 7) + 12, what is y when x = −10?
Explanation: y = (−10 − 7) + 12 = −17 + 12 = −5.
7. In n = 90 − 5(2 + m), what is n when m = 4?
Explanation: n = 90 − 5(2 + 4) = 90 − 5(6) = 90 − 30 = 60.
8. Each y in the table comes from y = 2x + 1. What y belongs to x = 12?
Explanation: y = 2(12) + 1 = 25.
3. Finding the Rule of a Function
Sometimes you are given a table and must find the rule. Look at how the outputs change as the inputs go up by 1. If y goes up by the same amount m each time, the rule has the form y = mx + b. Find b by using one input–output pair, or by reading the output where x = 0. Then check the rule on every column. To match a function to a table, substitute a few inputs into each choice and eliminate those that fail. A function with a constant rate of change (a straight line) can also be built from a word statement: a rate and a starting amount. For example, 2 books each week gives f(w) = 2w, and a $25 fee plus $4 per hour gives C = 25 + 4h.
Worked Example: Find the rule for x: 1, 2, 3, 4 and y: 2, 5, 8, 11.
y rises by 3 each time, so m = 3. At x = 1, 3(1) + b = 2, so b = −1. The rule is y = 3x − 1. Check: 3(4) − 1 = 11.
| x | 1 | 2 | 3 | 4 | 5 | … | 12 |
|---|---|---|---|---|---|---|---|
| y | 2 | 5 | 8 | 11 | 14 | … | ? |
1. Which function produces the table?
| x | 1 | 2 | 3 | 4 | 5 | … | 12 |
|---|---|---|---|---|---|---|---|
| y | 2 | 5 | 8 | 11 | 14 | … | ? |
Explanation: The outputs rise by 3 and x = 1 gives 2, so y = 3x − 1.
2. What number belongs in the shaded box (x = 12) of the table?
| x | 1 | 2 | 3 | 4 | 5 | … | 12 |
|---|---|---|---|---|---|---|---|
| y | 2 | 5 | 8 | 11 | 14 | … | ? |
Explanation: Using y = 3x − 1: y = 3(12) − 1 = 35.
3. Each y in this table comes from y = 3x + 5. What number belongs in the shaded box (x = 9)?
| x | −2 | −1 | 0 | 1 | 2 | … | 9 |
|---|---|---|---|---|---|---|---|
| y | −1 | 2 | 5 | 8 | 11 | … | ? |
Explanation: y = 3(9) + 5 = 32.
4. The price per scarf drops as more scarves are bought (see the table). Which function gives the price c for n scarves?
| Number of scarves (n) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Price per scarf (c) | $6.00 | $5.80 | $5.60 | $5.40 |
Explanation: At n = 1 the price is 6.00 and each extra scarf takes off 0.20, so c = 6.00 − 0.20(n − 1). Check n = 4: 6.00 − 0.60 = 5.40.
5. Which sequence of y-values could come from y = 4x − 3 for x = 1, 2, 3, 4, 5?
Explanation: y = 4(1) − 3 = 1, then 5, 9, 13, 17: it rises by 4 each time.
6. A reading program has a child read 2 books each week. Which function gives the books read after w weeks?
Explanation: Books = rate × weeks = 2w.
7. A plumber charges a $40 fee plus $30 per hour. Which function gives the cost C for h hours?
Explanation: A flat fee is added once (40) and the hourly rate is multiplied by the hours (30h).
4. Function Applications and Comparisons
Functions model work and money: pay, cost, profit, taxes, and more. Read the problem for the meaning of each variable, write or locate the function, substitute the given value, and compute carefully using the order of operations. Include the units in your answer. Functions are also used to compare options: evaluate each function at the same input and compare the results. If two options are graphed, the line that is lower at that input costs less, and the place where the lines cross is where the two options cost the same. Test tip: there is often more than one way to work a problem. Solve with the function, then use another approach (a quick estimate or the graph) to check.
Worked Example: Gym A charges P = 120 + 15m and Gym B charges P = 30m for m months. Which is cheaper for 10 months?
Gym A: 120 + 15(10) = 270. Gym B: 30(10) = 300. Gym A is cheaper by $30.
1. A job costs $2,000 per day for the first 7 days and $1,500 per day after that. For d > 7 the cost is C = 2000d + 1500(d − 7). What is the cost for 12 days?
Explanation: C = 2000(12) + 1500(5) = 24,000 + 7,500 = 31,500.
2. A company is paid $95,000 for a job with expenses of $5,400 per day. Profit is P = 95,000 − 5,400d. What is the profit if the job takes 14 days?
Explanation: P = 95,000 − 5,400(14) = 95,000 − 75,600 = 19,400.
3. A driver estimates travel time in hours as t = m ÷ 55, where m is miles. How long will a 330-mile trip take?
Explanation: t = 330/55 = 6.
4. Using t = m ÷ 55, how many hours will a 275-mile trip take?
Explanation: t = 275/55 = 5.
5. Alicia compares jobs for weekly pay P by hours h: Job 1: P = 9.50h; Job 2: P = 60 + 8.40h. If she works 30 hours, how much more does Job 2 pay than Job 1?
Explanation: Job 1: 9.50(30) = 285. Job 2: 60 + 8.40(30) = 312. Difference = 27.
6. With Job 3: P = 350 × (h ÷ 40), Alicia works 40 hours. Which is true?
Explanation: Job 1: 380. Job 2: 60 + 336 = 396. Job 3: 350. Job 2 pays the most.
7. Gym A: P = 120 + 15m. Gym B: P = 30m. For a 4-month stay, which is cheaper?
Explanation: A: 120 + 60 = 180. B: 30(4) = 120. Gym B is cheaper by $60.
8. Using the graph, which plan is cheaper for 300 minutes of calls, and what does it cost?
Explanation: At 300 minutes Plan A is 10 + 0.05(300) = 25 and Plan B is 0.10(300) = 30. Plan A is cheaper.
9. Using the graph, which plan is cheaper for 100 minutes of calls?
Explanation: At 100 minutes Plan A costs 15 and Plan B costs 10, so Plan B is cheaper.
10. At how many minutes do Plan A and Plan B cost the same?
Explanation: The lines cross at (200, 20): 10 + 0.05(200) = 20 and 0.10(200) = 20.
5. Function Notation
Function notation writes the output as f(x), read “f of x.” It does not mean f times x. For example, f(x) = 3x − 4 is the same rule as y = 3x − 4. The number inside the parentheses is the input: to find f(5), replace every x with 5. Other letters work the same way: g(t), C(d), f(w). A statement like f(2) = 7 says the point (2, 7) is on the graph. Sometimes you are given the output and must find the input: set the function equal to the output and solve. Test tip: the question always gives you enough information to figure out the function.
Worked Example: If f(x) = x2 + 3x, find f(−2).
f(−2) = (−2)2 + 3(−2) = 4 − 6 = −2.
1. If f(x) = 3x − 4, what is f(5)?
Explanation: f(5) = 3(5) − 4 = 11.
2. If f(x) = x2 + 3x, what is f(−2)?
Explanation: f(−2) = 4 − 6 = −2.
3. If g(x) = x2 − 2x + 1, what is g(−3)?
Explanation: g(−3) = 9 + 6 + 1 = 16.
4. If f(x) = |x − 6|, what is f(2)?
Explanation: f(2) = |2 − 6| = |−4| = 4.
5. If f(x) = 2x + 7 and f(a) = 15, what is a?
Explanation: 2a + 7 = 15, so 2a = 8 and a = 4.
6. For f(x) = 3x − 4, for which x does f(x) = 17?
Explanation: 3x − 4 = 17, so 3x = 21 and x = 7.
7. A baker makes 18 cookies per cup of oats. Which function gives the cookies f(c) from c cups?
Explanation: Cookies = 18 per cup times cups.
8. Using f(c) = 18c, how many cups of oats make 99 cookies?
Explanation: 18c = 99, so c = 99/18 = 5.5 cups.
9. If f(d) = 2000d + 1500(d − 7), what is f(10)?
Explanation: f(10) = 20,000 + 1500(3) = 24,500.
10. If f(4) = 9, which point is on the graph of f?
Explanation: The notation f(input) = output gives the point (input, output) = (4, 9).
6. Domain and Range
The domain of a function is the set of all input values that make sense. The range is the set of all output values the function can produce. To find the domain, look for values that must be excluded: division by zero is undefined, and a square root of a negative number is not a real number. Real-world situations can limit the domain too (time cannot be negative). To find the range, think about the smallest and largest outputs: a squared expression is never negative, an absolute value is never negative, and a negative sign in front flips the result. If you are unsure, graph the function or try several inputs and look at how the outputs behave.
Worked Example: Find the domain of f(x) = 11 − x2.
The denominator cannot be 0: 1 − x2 = 0 gives (1 + x)(1 − x) = 0, so x = ±1. The domain is all real numbers except −1 and 1.
1. What is the domain of f(x) = x − 2?
Explanation: Any number can be subtracted by 2, so the domain is all real numbers.
2. What is the domain of f(x) = 5x + 3?
Explanation: The denominator is 0 when x = −3, so x ≠ −3.
3. What is the domain of f(x) = √(x − 4)?
Explanation: The expression under the square root must be 0 or greater: x − 4 ≥ 0, so x ≥ 4.
4. What is the range of f(x) = x2?
Explanation: A squared real number is never negative, and 0 is reached at x = 0. The range is f(x) ≥ 0.
5. What is the range of f(x) = −|6x|?
Explanation: |6x| is 0 or positive, so its negative is 0 or negative.
6. What is the range of f(x) = x2 + 4x + 4?
Explanation: x2 + 4x + 4 = (x + 2)2, which is never negative and equals 0 at x = −2.
7. What is the range of f(x) = −(x2 − 1)?
Explanation: −(x2 − 1) = 1 − x2. Since x2 ≥ 0, the largest value is 1 (at x = 0).
8. A cost function is f(d) = 2000d + 1500(d − 7) for d ≥ 7 days. What is the smallest value in its range?
Explanation: At d = 7 the cost is 2000(7) + 0 = 14,000, and it only grows after that.
9. A road trip covers 100 miles before breakfast, then 60 miles per hour for h hours, with 0 ≤ h ≤ 8. Which gives the domain and range of f(h) = 100 + 60h?
Explanation: Hours h go from 0 to 8. At h = 0 the distance is 100 and at h = 8 it is 100 + 480 = 580.
10. What is the domain of f(z) = z2 − 9z2 − 6z + 9?
Explanation: The denominator z2 − 6z + 9 = (z − 3)2 is 0 only at z = 3.
Common Mistakes to Avoid
- Reading f(x) as f times x. It means the output of the function at input x.
- Forgetting parentheses when substituting a negative number: use x2 = (−3)2 = 9, not −32 = −9.
- Finding the next term of a pattern after checking only one pair of terms. Test the rule on all pairs.
- Mixing up the input and the output when a question gives f(x) = a number: solve for x.
- Using the wrong order of operations when evaluating, especially with parentheses and multiplication.
- Forgetting a flat fee or a starting amount in a real-world function.
- Forgetting to exclude values that make a denominator 0 or a square root negative when finding the domain.
- Mixing up domain (inputs) and range (outputs).
Frequently Asked Questions
What is a function?
A rule that gives exactly one output for each input.
What does f(x) mean?
It is function notation for the output of function f at the input x. It is read “f of x.”
How do I find the rule of a pattern?
Find the difference or the ratio between consecutive terms, test it on every pair, then apply it to continue the pattern.
How do I evaluate a function?
Replace the variable with the given input and follow the order of operations.
What are the domain and range?
The domain is the set of valid inputs. The range is the set of outputs the function can produce.
How do I find the domain?
Exclude any input that makes a denominator 0 or a square root of a negative number, and any value that does not make sense in the situation.
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.