Functions, Patterns and Sequences

This lesson trains you to evaluate functions written in f(x) notation, find the input that produces a given output, and decide whether a relation is a function. You will read domain and range from tables and graphs, build rules for arithmetic and geometric sequences, extend patterns, and handle composite and piecewise functions. These skills belong to ASVAB Mathematics Knowledge algebra, although no single question type is guaranteed.

A function is a rule that assigns each input exactly one output. Think of a vending machine: pressing button B4 should always drop the same snack. If one button could give two different snacks, the machine would not behave like a function. Inputs are usually called x, outputs y or f(x). Different inputs may share an output, but one input may never have two.

The notation $f(x)$ is read as f of x and means the output of f at input x; it is not multiplication. For

the value $f(2)$ means replace x by 2:

Fractions and decimals substitute the same way; just keep the arithmetic tidy. If

Put parentheses around every substituted value, especially negatives.

To find which input gives a certain output, set the rule equal to that output and solve. If

and the output is 22, then

so 5x = 25 and x = 5. Check:

A relation fails to be a function when one input is paired with two different outputs. The set {(2, 4), (3, 9), (2, 6)} is not a function because 2 maps to both 4 and 6. An identical pair listed twice, or several inputs sharing one output, as in {(1, 5), (2, 5)}, is still a function.

The domain is the set of all allowed inputs, and the range is the set of outputs actually produced. For the pairs (1, 2), (3, 2) and (5, 8), the domain is {1, 3, 5} and the range is {2, 8}. On a graph, the domain is the horizontal spread and the range is the vertical spread.

Some inputs are banned because they cause impossible arithmetic. A denominator cannot be zero, so

excludes

In real numbers, a square root needs a nonnegative inside, so $\sqrt{x-4}$ requires

The same function can appear as a rule, a table, a list of ordered pairs, or a graph. A table row with x = 2 and y = 7 is the ordered pair (2, 7) and one plotted point. To read f(4) from a graph, find 4 on the horizontal axis, move straight up or down to the curve, and read its height.

To test whether a graph shows a function, imagine sliding a vertical line across it. If any vertical line touches the graph more than once, some input has two outputs, so the graph is not a function. A parabola opening upward passes; a circle fails, since a vertical line through its middle hits it twice.

A linear function has the form f(x) = mx + b. Each time x increases by 1, the output changes by the same amount m, the slope. The constant b is the output when x = 0. For

the outputs for x = 0, 1, 2, 3 are 7, 5, 3, 1, dropping by 2 each step.

Rate of change compares the change in output with the change in input:

From (1, 4) to (4, 13), the output rises 9 while the input rises 3, so the rate is 3 per unit.

Word problems often hide a function. A gym charges a 25-dollar sign-up fee plus 30 dollars per month, so the cost is

Here 30 is the rate per month and 25 is the starting value. After 6 months,

An arithmetic sequence adds the same number, the common difference d, to get each new term. In 9, 13, 17, 21, ..., d = 4, so the next term is 25. A negative d makes the terms fall, as in 20, 17, 14, 11, where

The nth term is

For 5, 8, 11, ..., the 20th term is

Use n - 1, not n, because reaching the first term takes no steps.

You can run the formula backward. A sequence starts at 3 with d = 5. Which term is 48? Solve

so n = 10.

To add the first n terms, average the first and last terms and multiply by n:

For 2, 5, 8, ..., 20, which has seven terms, the sum is

A geometric sequence multiplies by the same number, the common ratio r, each time. Find r by dividing a term by the one before it. In 5, 15, 45, 135, ...,

so the next term is 405. When r is between 0 and 1, the terms shrink.

The nth term of a geometric sequence is

For 4, 8, 16, ..., r = 2, so the 7th term is

Repeated multiplication models growth and decay. A town of 2,000 people growing 10% per year multiplies by 1.1 each year: 2,200, then 2,420. A car worth 20,000 dollars that loses 20% per year keeps 80% each time: 16,000, then 12,800 dollars. Growth uses r greater than 1; decay uses r between 0 and 1.

If the gaps between terms are not constant but the gaps of those gaps are, the pattern is quadratic. In 1, 4, 9, 16, ..., the gaps 3, 5, 7 rise by 2, and the terms are $n^2$. A pattern such as 3, 6, 11, 18, ... follows $n^2+2$, so the next term is 27.

A recursive rule gives a starting term and says how to get each term from the one before. With

the terms are 3, then 7, then 19, then 55. You must build the terms one at a time, in order.

Some patterns switch between two rules or weave two sequences together. In 2, 5, 4, 7, 6, 9, ..., the steps alternate between adding 3 and subtracting 1, so the next term is 8. In 1, 10, 2, 20, 3, 30, ..., odd positions count 1, 2, 3 while even positions count by tens, so the next term is 4.

$f(g(x))$ means apply g first, then feed that result into f. With

and g(x) = 2x, g(4) = 8 and f(8) = 13. Order matters:

A piecewise function uses different rules on different parts of the domain. Suppose

when

when

and r(2) = 6. At the boundary x = 1, the second rule applies, so r(1) = 3.

Many word problems are sequences in disguise. A runner trains 2 miles the first week and adds half a mile each week, so the weekly distance is arithmetic with first term 2 and d = 0.5. Week 9 is

Doubling, halving, or a fixed percent change signals a geometric pattern instead.

When the inputs in a table do not rise by 1, divide each output change by its input change. If x goes 0, 2, 6 and y goes 1, 7, 19, the changes are 6 over 2 and 12 over 4, both 3, so

Check answer choices instead of recomputing from scratch. If

and a choice claims x = 6 gives an output of 19, compute 16 and reject it; x = 7 gives 19. For sequences, confirm the pattern with two consecutive gaps or ratios before extending it. Watch signs: an even power of a negative number is positive.

Problem: Find $h(-3)$ for

Step 1: Replace every x with (−3). Always use parentheses around a negative input.

Step 2: Do the exponent first. (−3)² =

because a negative times a negative is positive.

Step 3: Multiply:

Subtracting a negative is the same as adding, so −(−3) = +3.

Step 4: Add from left to right:

Answer: 25.

Check: A common mistake is writing −3² = −9, which would give

Squaring a negative number always gives a positive result, so the term 2x² must be positive: 18. Then

is right.

Problem: A table gives the points (1, 2), (3, 8) and (5, 14). Find the linear rule y = mx + b.

Step 1: Find the slope. From (1, 2) to (3, 8), x goes up

and y goes up

Step 2: Find b. Put the slope and the point (1, 2) into y = mx + b.

Step 3: Subtract 3 from both sides:

Answer:

Check: Test all three points. x = 1:

x = 3:

x = 5:

All three match the table.

Problem: For the sequence 6, 10, 14, ..., find the 15th term and the sum of the first 15 terms.

Step 1: Find the common difference:

so d = 4. The first term is 6.

Step 2: To reach the 15th term, start at the 1st term and add d fourteen times (

).

Multiply first:

Step 3: For the sum, use: number of terms × (first + last) ÷ 2. First add

Step 4: Halve 68 first to keep the numbers small:

Answer: The 15th term is 62, and the sum is 510.

Check: The average term is

and 15 terms at an average of 34 make 510.

Problem: A culture of 300 bacteria doubles every hour. How many are there after 5 hours?

Step 1: Doubling means multiplying by 2 each hour. After 5 hours you multiply by 2 five times, which is $2^5$.

Step 2: Work out the power:

Step 3: Multiply:

Answer: 9,600 bacteria.

Check: Track it hour by hour: 300, then 600, 1,200, 2,400, 4,800, and 9,600 after hour 5.

Problem: With

find $p(q(1))$.

Step 1: Work from the inside out. First find q(1) by putting 1 into q.

Step 2: The output of q becomes the input of p. So p(q(1)) = p(5).

Step 3: Square 5:

Then subtract 1.

Answer: 24.

Check: Be careful not to work outside in.

and q(0) = 2, which is q(p(1)), a different question. For p(q(1)), q always goes first: q(1) = 5, p(5) = 24.

Problem: Find the next term of 4, 7, 12, 19, 28, ...

Step 1: Find the gaps between neighboring terms:

Step 2: The gaps 3, 5, 7, 9 go up by 2 each time, so the next gap is

Step 3: Add the next gap to the last term.

Step 4: Find a rule as a second method. Compare each term with the square numbers 1, 4, 9, 16, 25: every term is 3 more (

and so on). So the nth term is $n^2+3$.

The next term is term number 6.

Answer: 39.

Check: Both methods give 39. Also check the rule on term 5: 5² +

which matches the sequence.

Problem: Let

for

for

Find $s(2)+s(4)$.

Step 1: Decide which rule to use for x = 2. The first rule covers x ≤ 2, which includes 2 itself. So use 5 − x.

Step 2: For x = 4, the number 4 is greater than 2, so use the second rule, 2x.

Step 3: Add the two results.

Answer: 11.

Check: The boundary is the tricky part. If you wrongly used 2x for x = 2, you would get 4 instead of 3 and a total of 12. Since ≤ includes 2, the first rule is correct and the total is 11.

Problem: Which inputs are excluded from the domain of

Step 1: You can never divide by zero. So find the x values that make the denominator zero.

Step 2: Add 9 to both sides.

Step 3: Which numbers squared give 9? Both

Step 4: Think about the numerator. It is zero when

but a zero on top is fine: 0 divided by a nonzero number is just 0.

Answer: x = 3 and

are excluded. Every other real number is allowed.

Check: At x = 3 the bottom is

and at

it is

so both fail. At

the value is

which is fine.

Treating $f(x)$ as f times x leads nowhere; it names an output. Dropping parentheses around negatives turns

into -9. Using n instead of n - 1 in the arithmetic formula overshoots by one step, which is why wrong choices often differ from the key by exactly d. Mixing up arithmetic and geometric patterns is avoided by checking whether gaps or ratios stay constant. Work compositions inside out, read piecewise boundary signs carefully, and remember that a repeated identical pair never breaks a function.

A function gives each input one output; use the vertical line test on graphs and look for one input with two outputs in lists. Evaluate by substituting in parentheses, and find an input by setting the rule equal to the output. Domains exclude zero denominators and negative square-root insides. Arithmetic sequences add d, with

Geometric sequences multiply by r, with

Check second differences for quadratic patterns, and work compositions inside out.