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College Algebra

Parent Functions

Lesson

A handful of parent functions show up again and again in algebra. Each is the simplest member of a family — every other function in the family is built by transforming one of these. Learning their shapes by heart makes everything that follows easier.

The six parent functions:

  • Linear (identity): f(x)=xf(x) = x — straight line through the origin. Domain and range: all real numbers.
  • Quadratic: f(x)=x2f(x) = x^2 — U-shape (parabola), vertex at the origin. Range: [0, ∞)[0,\,\infty).
  • Cubic: f(x)=x3f(x) = x^3 — S-shape through the origin. Domain and range: all real numbers.
  • Square root: f(x)=xf(x) = \sqrt{x} — starts at the origin and curves up-right. Domain and range: [0, ∞)[0,\,\infty).
  • Absolute value: f(x)=∣x∣f(x) = |x| — V-shape with the vertex at the origin. Range: [0, ∞)[0,\,\infty).
  • Reciprocal: f(x)=1xf(x) = \dfrac{1}{x} — two hyperbolic branches. Domain and range: all real numbers except 0.

For each, evaluating at a point means substituting that value of xx into the rule.

Worked example 1

f(x)=x,f(49)=?f(x) = \sqrt{x}, \quad f(49) = ?

Substitute x=49x = 49:

f(49)=49=7f(49) = \sqrt{49} = 7

Worked example 2

f(x)=1x,f ⁣(−13)=?f(x) = \frac{1}{x}, \quad f\!\left(-\tfrac{1}{3}\right) = ?

Substitute x=−13x = -\tfrac{1}{3}:

f ⁣(−13)=1−1/3=−3f\!\left(-\tfrac{1}{3}\right) = \frac{1}{-1/3} = -3

Dividing 1 by a fraction flips it.

How to type your answer

Type a single number. Negatives use a minus sign; fractions use a slash. Examples: 7, -27, 1/2, -1/4.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

f(x)=x2; f(4)=?f(x) = x^2;\ f(4) = ?

Problem 2

f(x)=x3; f(2)=?f(x) = x^3;\ f(2) = ?

Problem 3

f(x)=x; f(9)=?f(x) = \sqrt{x};\ f(9) = ?

Problem 4

f(x)=∣x∣; f(−7)=?f(x) = |x|;\ f(-7) = ?

Practice

Standard problems matching the lesson.

Problem 5

f(x)=x2; f(−5)=?f(x) = x^2;\ f(-5) = ?

Problem 6

f(x)=x3; f(−3)=?f(x) = x^3;\ f(-3) = ?

Problem 7

f(x)=x; f(25)=?f(x) = \sqrt{x};\ f(25) = ?

Problem 8

f(x)=1x; f(2)=?f(x) = \dfrac{1}{x};\ f(2) = ?

Problem 9

f(x)=1x; f(−4)=?f(x) = \dfrac{1}{x};\ f(-4) = ?

Problem 10

f(x)=∣x∣; f(−12)=?f(x) = |x|;\ f(-12) = ?

Problem 11

f(x)=x3; f(−4)=?f(x) = x^3;\ f(-4) = ?

Problem 12

f(x)=x; f(0)=?f(x) = \sqrt{x};\ f(0) = ?

Problem 13

f(x)=x2; f(−6)=?f(x) = x^2;\ f(-6) = ?

Problem 14

f(x)=1x; f(5)=?f(x) = \dfrac{1}{x};\ f(5) = ?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

f(x)=x; f(144)=?f(x) = \sqrt{x};\ f(144) = ?

Problem 16

f(x)=x3; f(−10)=?f(x) = x^3;\ f(-10) = ?

Problem 17

f(x)=1x; f ⁣(14)=?f(x) = \dfrac{1}{x};\ f\!\left(\tfrac{1}{4}\right) = ?

Problem 18

f(x)=1x; f ⁣(−12)=?f(x) = \dfrac{1}{x};\ f\!\left(-\tfrac{1}{2}\right) = ?

Problem 19

f(x)=x; f ⁣(14)=?f(x) = \sqrt{x};\ f\!\left(\tfrac{1}{4}\right) = ?

Problem 20

f(x)=x2; f ⁣(−13)=?f(x) = x^2;\ f\!\left(-\tfrac{1}{3}\right) = ?

Problem 21

f(x)=x3; f ⁣(12)=?f(x) = x^3;\ f\!\left(\tfrac{1}{2}\right) = ?

Problem 22

f(x)=∣x∣; f ⁣(−34)=?f(x) = |x|;\ f\!\left(-\tfrac{3}{4}\right) = ?