Lecture Notes on Functions and Inverses

Key Topics Covered

1. Calculating the Value of Functions
2. Finding Inverse Functions
3. Solving for Values of Variables in Functions
4. Composite Functions
5. Graphical Analysis of Functions
6. Domain and Range of Functions

1. Calculating the Value of Functions

• Problem: Calculate f(2) for the function f(x) = x^3 + 5x^2 - 9.
• Solution:
  • Substitute x with 2: 2^3 + 5(2^2) - 9
  • Compute: 8 + 20 - 9 = 19
  • Answer: 19

2. Finding Inverse Functions

• Problem: Find the inverse of f(x).
• Solution:
  • Replace f(x) with y: y = f(x)
  • Swap x and y: x = f(y)
  • Solve for y:
    • Raise both sides to the fifth power
    • Simplify: x^5 = 7y - 3
    • Isolate y: y = (x^5 + 3) / 7
  • Answer: The inverse function is (x^5 + 3) / 7

3. Solving for Values in Functions

• Problem: If f(x) = -15, find x.
• Solution:
  • Set f(x) to -15 and solve the equation 2x^2 - 5x - 3 = -15
  • Rearrange and factor: 2x^2 - 5x + 12 = 0
  • Solve using factoring techniques:
    • Factorize: (2x + 3)(x - 4) = 0
    • Solve for x:
      • x = -3/2
      • x = 4
  • Valid Answer: x = 4

4. Composite Functions

• Problem: Find f(g(x)), given g(x) = x - 5 and f(x) = x^2 + 3x + 2.
• Solution:
  • Substitute g(x) into f(x): f(x-5)
  • Simplify: (x - 5)^2 + 3(x - 5) + 2
  • Compute: x^2 - 10x + 25 + 3x - 15 + 2
  • Combine like terms: x^2 - 7x + 12
  • Answer: x^2 - 7x + 12

5. Graphical Analysis of Functions

• Problem: Determine if graphs are functions and if they are one-to-one.
• Solution:
  • Vertical Line Test for functions:
    • If a vertical line touches the graph more than once, it's not a function.
    • Example: Graph C fails; touches vertical line multiple times.
  • Horizontal Line Test for one-to-one functions:
    • If a horizontal line touches the graph more than once, it's not one-to-one.
    • Example: Graph C fails; touches horizontal line multiple times.
  • Answers:
    • Graph not a function: C
    • Graph not one-to-one: C

6. Domain and Range of Functions

• Problem: Find the domain of f/g.
• Solution:
  • Given functions: f(x) = 7x - 3, g(x) = x^2 + 2x - 15
  • Domain restricted by zeros in the denominator: x^2 + 2x - 15 ≠ 0
  • Factor the quadratic: (x + 5)(x - 3) ≠ 0
  • Solve: x ≠ -5 and x ≠ 3
  • Domain: (-∞, -5) U (-5, 3) U (3, ∞)