Lecture Notes on Functions and Inverses

Key Topics Covered

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

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

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

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

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

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, ∞)