Function notation and evaluating functions
Read f(x) as a rule that turns each input into one output, evaluate a function at given values, and solve f(x) = k.
What a learner can do afterwards
- Work out f(3) and f(-2) for a given rule
- Solve f(x) = 0 for a quadratic rule
- Explain why each input is allowed only one output
1 · Read
An nth-term rule takes a position number as input and gives a term as output. A function also gives exactly one output for each allowed input. We name a function f and write its rule, for example f(x) = 3x - 2. The x in f(x) is a placeholder for the input. Writing f(4) means replace x with 4 in the rule and calculate the output.
Evaluate by replacing every x with the input. For f(x) = 3x - 2, f(-1) = 3(-1) - 2 = -5. Here 3(-1) means multiplication, not 3 - 1. Different inputs may share an output: 3 and -3 both give 5 under x² - 4. Each allowed input still has exactly one output.
To solve f(x) = k, find the input or inputs that give output k. Set the rule equal to k and solve the equation. When k = 0, these inputs are the roots. A linear rule with nonzero slope and all real inputs gives one input for every real output. A quadratic may give two real inputs, one at its turning point, or none when k is outside its range.
For f(x) = 3x - 2, evaluate three inputs. f(4) = 3(4) - 2 = 10. f(-1) = 3(-1) - 2 = -5. f(0) = 3(0) - 2 = -2. Keep negative inputs in brackets. In a polynomial rule, substituting zero removes every term containing x, leaving the constant term.
For g(x) = x² - 4, g(3) = 9 - 4 = 5 and g(-3) = (-3)² - 4 = 9 - 4 = 5. The whole negative input is squared: (-3)(-3) = 9. Writing -9 - 4 would square the magnitude and then apply a minus sign, which is not this substitution. Also g(2) = 4 - 4 = 0. Opposite inputs may share an output without breaking the one-output rule.
For h(x) = x² - 5x + 6, find inputs giving 0: x² - 5x + 6 = 0 factors as (x - 2)(x - 3) = 0. The inputs are 2 and 3. Check h(2) = 4 - 10 + 6 = 0. For output 2, subtract 2 from the rule: x² - 5x + 4 = 0. This factors as (x - 1)(x - 4) = 0, giving inputs 1 and 4. In each case, set the rule equal to the requested output.
An input may be an expression. For p(t) = 4t - 3, find p(x + 2) by replacing every t with the whole expression x + 2: 4(x + 2) - 3 = 4x + 5. Keep brackets during substitution so the coefficient multiplies every term in the input.
To recover p(x) = ax + b from two values, turn each value into an equation. If p(2) = 5 and p(3) = 8, then 2a + b = 5 and 3a + b = 8. Subtract to get a = 3, then use 2a + b = 5 to find b = -1. Now p(5) = 3(5) - 1 = 14.
Count distinct inputs when solving a quadratic rule. The equation (x - 7)² = 0 has only input 7, since both factors give that same value. A square is nonnegative for real inputs. Thus (x + 6)² + 2 is always at least 2 and never gives output 0. Expand a squared bracket to check whether a quadratic can be written this way.
Read f(x) as a rule with a placeholder, evaluate by substituting the input with care for negative numbers, and solve f(x) = k by setting the rule equal to k and solving for the inputs.
The practice questions and readings were written for this dataset. The referenced video is by Khan Academy.
More to read
Reading a function's name
In p(6), p names the function and 6 names the input. The notation asks for an output, not the product of p and 6.
For p(x) = 4x - 7, input 3 gives 4(3) - 7 = 5. Write the substituted expression before calculating, especially when the input is negative.
An allowed input has one output. The assignments (8, 2) and (8, 5) cannot both belong to the same function. Assignments (8, 2) and (9, 2) can: different inputs may share an output.
Finding inputs from an output
Evaluating starts with an input. Solving p(x) = k starts with the output k.
For p(x) = x² - 7x + 10, find the inputs giving zero. The equation factors as (x - 2)(x - 5) = 0, so the inputs are 2 and 5. Each input still gives one output; both happen to give zero.
For a linear rule such as q(x) = 6x + 1, an output of 19 gives 6x + 1 = 19, hence x = 3.
Checking substitution
A useful check is to write out the substituted expression and compare its value with the result of an equivalent calculation.
For p(x) = x² - 6x, input 8 gives 64 - 48 = 16. The equivalent rule x(x - 6) gives 8(8 - 6) = 16 too. Agreement checks this evaluation; it does not by itself prove that two arbitrary rules are identical.
Applying p again is a new evaluation, not an undoing. Here p(16) = 256 - 96 = 160, which is different from 16. Do not expect feeding an output through the same function to return it unchanged.
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2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
28 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.