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Inverse Functions in Precalculus

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  • Steps to find the inverse of a function
    1. Replace f(x) with y.
    2. Interchange x and y.
    3. Solve for y.
    4. Replace y by f-1(x) if the inverse exists.
  • How to verify if two functions are inverses
    Check if f(f-1(x)) = x and f-1(f(x)) = x.
  • What does it mean if a function passes the Horizontal Line Test?
    The function is one-to-one and has an inverse that is also a function.
  • Inverse of the function \(f(x) = 7x + 6\)
    The inverse is \(f^{-1}(x) = \frac{x - 6}{7}\).
  • How to find the inverse of \(f(x) = 3x - 1\)
    Replace f(x) with y, interchange x and y, solve for y: \(f^{-1}(x) = \frac{x + 1}{3}\).
  • What is the inverse of \(f(x) = 5x + 1\)?
    The inverse is \(f^{-1}(x) = \frac{x - 1}{5}\).
  • Why does \(f(x) = x^2\) not have an inverse function?
    It fails the Horizontal Line Test, so it is not one-to-one and does not have an inverse function.
  • What is the graphical relationship between a function and its inverse?
    The graph of the inverse is the reflection of the function's graph about the line \(y = x\).
  • How to verify the inverse function algebraically?
    Show that f(f-1(x)) = x and f-1(f(x)) = x by substitution and simplification.
  • Inverse of \(f(x) = \frac{x + 4}{x - 2}\)
    Solve for y: \(f^{-1}(x) = \frac{2x + 4}{x - 1}\).
  • What does it mean if a function is one-to-one?
    Each output corresponds to exactly one input, so the function has an inverse that is also a function.
  • How to find the inverse of \(f(x) = -8x + 8\)
    The inverse is \(f^{-1}(x) = -\frac{x - 8}{8}\).
  • What is the inverse of \(f(x) = 3x - 7\)?
    The inverse is \(f^{-1}(x) = \frac{x + 7}{3}\).
  • How to use the Horizontal Line Test
    Draw horizontal lines through the graph; if any line intersects more than once, the function is not one-to-one.
  • What is the inverse of \(f(x) = 3x - 1\)?
    The inverse is \(f^{-1}(x) = \frac{x + 1}{3}\).
  • How to find the inverse of a function algebraically
    Replace f(x) with y, interchange x and y, then solve for y.
  • What is the inverse of \(f(x) = x - 4\)?
    The inverse is \(f^{-1}(x) = x + 4\).
  • What does the notation f-1(x) represent?
    It represents the inverse function of f(x), not the reciprocal.
  • How to verify if g(x) is the inverse of f(x)
    Check if f(g(x)) = x and g(f(x)) = x for all x in the domain.
  • What is the inverse of \(f(x) = x^3\)?
    The inverse is \(f^{-1}(x) = \sqrt[3]{x}\).