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Formulas and f(x)
Some questions hand you a formula and ask you to use it; others ask you to rearrange it; others write it as f(x). All three are the same skill as solving an equation, with letters where the numbers used to be.
A formula is an equation you can rearrange
- A = l × w
- A ÷ l = l × w ÷ l
- w = A ÷ l
A formula ties several quantities together: area, length and width, for example. You can plug in numbers for all but one letter and solve, or rearrange it first so the letter you want stands alone.
f(x) is a machine
f(x) = 3x − 2 means: whatever goes in as x, multiply it by 3 and subtract 2. f(5) is what comes out when 5 goes in: 3 × 5 − 2 = 13. It is not f times x.
How to solve them
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1
Plug numbers into a formula
Replace each letter with its number (in parentheses if it is negative) and work it out with the order of operations.
Next step: Solve for a letterExampleF = 1.8C + 32. What is 25 °C in Fahrenheit?
- Replace C with 25: F = 1.8 × 25 + 32.
- 1.8 × 25 = 45.
- 45 + 32 = 77.
Answer 77 °F
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2
Solve a formula for one letter
Treat every other letter as if it were a number. Undo the operations around the letter you want, in reverse order, on both sides.
Next step: Plug in after- P = 2l + 2w
- P − 2w = 2l
- l = (P − 2w) ÷ 2
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3
Rearrange first, then plug in
When the unknown is not the letter alone on the left, it is often cleaner to rearrange first and only then put the numbers in.
C = 2πr → r = C ÷ (2π)
Next step: f(x)ExampleA circle has a circumference of 31.4. Use π ≈ 3.14 to find its radius.
- r = C ÷ (2π) = 31.4 ÷ (2 × 3.14).
- 2 × 3.14 = 6.28.
- 31.4 ÷ 6.28 = 5.
Answer r = 5
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4
f(x) notation: replace every x
Find every x in the rule and put the input in its place. Then calculate.
Next step: Negative inputs- f(x) = 3x² − 2x + 1, find f(4)
- 3(4)² − 2(4) + 1
- 3 × 16 − 8 + 1
- 48 − 8 + 1
- 41
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5
Negative inputs: always use parentheses
A negative input goes in wrapped in parentheses, so that squaring it gives a positive number and subtracting it turns into adding.
Next step: f of an expression- f(x) = x² − 5x, find f(−2)
- (−2)² − 5(−2)
- 4 + 10
- 14
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6
f(a + 3): the whole expression goes in
If the input is an expression, replace x with the whole expression in parentheses, then simplify.
- f(x) = 2x + 7, find f(a + 3)
- 2(a + 3) + 7
- 2a + 6 + 7
- 2a + 13
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Squaring a negative input without parentheses
f(x) = x², f(−3):
Wrong−3² = −9
Right(−3)² = 9
Dividing only one term
Solve y = 4x + 8 for x:
Wrongx = y ÷ 4 + 8
Rightx = (y − 8) ÷ 4
Reading f(x) as f times x
f(5):
Wrongf × 5
Rightthe output when x = 5
Replacing only one x
f(x) = x² + x, f(3):
Wrong9 + x
Right9 + 3 = 12
5 questions at test level
Pick an answer, then read why. No timer, no account.
If f(x) = 5x − 3, what is f(4)?
Show the answer
A. Replace x with 4: 5 × 4 − 3 = 20 − 3 = 17. 20 forgets the − 3, and 23 adds it. f(4) is 17.
The perimeter of a rectangle is P = 2l + 2w. If P = 46 and w = 8, what is l?
Show the answer
B. l = (P − 2w) ÷ 2 = (46 − 16) ÷ 2 = 30 ÷ 2 = 15. 30 forgets to divide by 2, and 19 subtracts only one width. l is 15.
Solve the formula A = ½bh for h.
Show the answer
B. Multiply both sides by 2: 2A = bh. Divide both sides by b: h = 2A ÷ b. Check with b = 4 and h = 5: A = 10, and 2 × 10 ÷ 4 = 5. The answer is h = 2A ÷ b.
If g(x) = x² − 4x + 2, what is g(−3)?
Show the answer
D. Use parentheses: (−3)² − 4(−3) + 2 = 9 + 12 + 2 = 23. −1 loses the sign of −4 × (−3), and 5 squares −3 to −9. g(−3) is 23.
Use F = 1.8C + 32 to convert 35 °C to degrees Fahrenheit.
Show the answer
C. 1.8 × 35 = 63, and 63 + 32 = 95. 63 forgets the + 32, and 120.6 adds 32 to 35 before multiplying. 35 °C is 95 °F.
9 flashcards
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All 9 cards known.
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