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Preparation for Calculus

P.3 Functions and Their Graphs

Focus on domain restrictions, composition, transformations, even and odd functions.

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Ideas in this lesson

01Domain restrictionsWorked steps · five questions02CompositionWorked steps · five questions03TransformationsWorked steps · five questions04Even and odd functionsWorked steps · five questions
Idea 1 of 4

Domain restrictions

Require real radicands to be nonnegative and denominators to be nonzero. Enforce both restrictions together.

Worked example 1

Find the domain of f(x)=x−2f(x) = \sqrt{x - 2}.
  1. x−2≥0x - 2 \ge 0
  2. x≥2x \ge 2
  3. D=[2,∞)D = [2, \infty )

Answer: [2,∞)[2, \infty )

Worked example 2

Find the domain of f(x)=1x−3f(x) = \dfrac{1}{x - 3}.
  1. x−3≠0x - 3 \ne 0
  2. x≠3x \ne 3
  3. D=D = ℝ ∖ {3}\{3\}

Answer: All real numbers except 33

Idea 2 of 4

Composition

Apply the inner function first. Composition order matters, and the inner output must belong to the outer domain.

Worked example 1

Let f(x)=x2f(x) = x^{2} and g(x)=x+2g(x) = x + 2. Find f(g(1))f(g(1)).
  1. g(1)=1+2=3g(1) = 1 + 2 = 3
  2. f(g(1))=(3)2=9f(g(1)) = (3)^{2} = 9

Answer: 99

Worked example 2

Let f(x)=x2f(x) = x^{2} and g(x)=x+3g(x) = x + 3. Find g(f(2))g(f(2)).
  1. f(2)=22=4f(2) = 2^{2} = 4
  2. g(f(2))=4+3=7g(f(2)) = 4 + 3 = 7

Answer: 77

Idea 3 of 4

Transformations

Horizontal changes act inside the rule; vertical changes act outside. Identify the parent function before sketching.

Worked example 1

For f(x)=(x−2)2+1f(x) = (x - 2)^{2} + 1, give the vertex xx-coordinate.
  1. (x−2)2≥0(x - 2)^{2} \ge 0; (x−2)2=0(x - 2)^{2} = 0 ⇔ x=2x = 2
  2. Vertex =(2,1)= (2, 1)
  3. f(x)≥1f(x) \ge 1; requested value =2= 2

Answer: 22

Worked example 2

For f(x)=(x−3)2+2f(x) = (x - 3)^{2} + 2, give the minimum value.
  1. (x−3)2≥0(x - 3)^{2} \ge 0; (x−3)2=0(x - 3)^{2} = 0 ⇔ x=3x = 3
  2. Vertex =(3,2)= (3, 2)
  3. f(x)≥2f(x) \ge 2; requested value =2= 2

Answer: 22

Idea 4 of 4

Even and odd functions

Use an algebraic symmetry test, rather than a few sampled points, to classify a function.

Worked example 1

Classify f(x)=x4+2f(x) = x^{4} + 2 as even, odd, or neither.
Coordinate graph-2-1.5-1-0.50.511.525101520xy
  1. f(−x)=x4+2f(-x) = x^{4} + 2
  2. f(−x)=f(x)f(-x) = f(x)
  3. f(−x)=f(x)  ⟹  f(-x) = f(x) \implies Even

Answer: Even

Worked example 2

Classify f(x)=x3−3xf(x) = x^{3} - 3x as even, odd, or neither.
Coordinate graph-2-1.5-1-0.50.511.525101520xy
  1. f(−x)=−x3+3xf(-x) = -x^{3} + 3x
  2. f(−x)=−f(x)f(-x) = -f(x)
  3. f(−x)=−f(x)  ⟹  f(-x) = -f(x) \implies Odd

Answer: Odd

Resource alignment: Larson & Edwards, Calculus of a Single Variable, Tenth Edition, §P.3,pP.3, p. 1919. Explanations, diagrams and numerical practice variants are original MathLab content aligned to this section.