2.3 Composing functions and finding inverses

• Function composition involves first evaluating \( f(x) \) and then using the result as the input for \( g(x) \).

  False
• Steps to compose functions:

  1️⃣ Identify the functions \( f(x) \) and \( g(x) \)

  2️⃣ Substitute \( g(x) \) into \( f(x) \)

  3️⃣ Simplify the expression
• If \( f(x) = x^2 + 3 \) and \( g(x) = 2x - 1 \), what is \( f(g(x)) \)?

  \( 4x^2 - 4x + 4 \)
• What is the composition \( f(g(x)) \) given \( f(x) = x^2 + 3 \) and \( g(x) = 2x - 1 \)?

  $4x^{2} - 4x +$$4$
• Composing functions involves applying one function to the output of another.


  True
• What condition must a function \( f(x) \) satisfy for its inverse to exist?

  One-to-one
• The composition \( f(f^{-1}(x)) = x \) for all \( x \) in the domain of \( f \).


  True
• The inverse function f^-1(x) reverses the operation of f(x).


  True
• Another condition to verify that f(x) and f^-1(x) are inverses is f^-1(f(x)) = x
• Steps to compose two functions f(x) and g(x) to find f(g(x))

  1️⃣ Identify f(x) and g(x)

  2️⃣ Substitute g(x) into f(x)

  3️⃣ Simplify the resulting expression
• The inverse function f^-1(x) reverses the operation of f(x).

  True
• What does function composition involve?

  Applying one function to another
• What is the notation for function composition when applying \( g(x) \) first and then \( f(x) \)?

  \( f(g(x)) \) or \( f \circ g(x) \)
• If \( f(x) = x^2 + 3 \) and \( g(x) = 2x - 1 \), what is \( f(g(x)) \)?

  \( 4x^2 - 4x + 4 \)
• Steps to compose functions:

  1️⃣ Identify the functions \( f(x) \) and \( g(x) \)

  2️⃣ Substitute \( g(x) \) into \( f(x) \)

  3️⃣ Simplify the resulting expression
• Steps to compose functions

  1️⃣ Identify the functions \( f(x) \) and \( g(x) \) ||| Substitute \( g(x) \) into \( f(x) \) ||| Simplify the resulting expression
• What is the inverse function of \( f(x) = 2x + 3 \)?

  $\frac{x - 3}{2}$
• What line do the graphs of \( f(x) \) and \( f^{-1}(x) \) reflect across?

  $y =$$x$
• For f^-1(x) to exist, f(x) must be one-to-one
• What test must f(x) pass to be one-to-one?

  Horizontal line test
• What is the notation for function composition?

  f(g(x)) or f ∘ g(x)
• If f(x) = x^2 + 3 and g(x) = 2x - 1, then f(g(x)) = (2x - 1)^2 + 3
• For the inverse function f^-1(x) to exist, the original function f(x) must pass the horizontal line test.
• Function composition involves applying one function to the result of another.
• The parenthetical notation f(g(x)) and the circle notation f ∘ g(x) both mean the same thing in function composition.


  True
• If \( f(x) = x^2 \) and \( g(x) = 2x + 1 \), then \( f(g(x)) = 4x^2 + 4x + 1 \)


  True
• Generally, \( f(g(x)) = g(f(x)) \)

  False
• The inverse function \( f^{-1}(x) \) reverses the operation of \( f(x) \)


  True
• Steps to find the inverse of a function \( f(x) \) algebraically

  1️⃣ Replace \( f(x) \) with \( y \)

  2️⃣ Swap \( x \) and \( y \)

  3️⃣ Solve for \( y \)

  4️⃣ Replace \( y \) with \( f^{-1}(x) \)
• If \( f(x) = x^2 + 3 \) and \( g(x) = 2x - 1 \), then \( f(g(x)) = 4x^2 - 4x + 4
• When composing functions, the notation \( f(g(x)) \) means that \( g(x) \) is evaluated first
• The notation for function composition includes both \( f(g(x)) \) and \( f \circ g(x) \).


  True
• The inverse of a function always exists for any function.

  False
• The notation \( f(g(x)) \) indicates that \( g(x) \) is evaluated first
• The inverse function of \( f(x) \) is denoted as f^{-1}(x)
• To verify that \( f(x) \) and \( f^{-1}(x) \) are inverses, we check that \( f(f^{-1}(x)) \) equals x
• What is the notation for the inverse function of f(x)?

  f^-1(x)
• To verify that f(x) and f^-1(x) are inverses, we check that f(f^-1(x)) = x.


  True
• In function composition f(g(x)), we first evaluate g(x) and then use the result as input for f(x).


  True
• If f(g(x)) = (2x - 1)^2 + 3, what is the simplified form?

  4x^2 - 4x + 4