Inverse Functions - Quick Review

Inverse Functions: Quick Recall

  • Key idea: If f and g are inverses, their compositions yield the identity.

    • f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x
    • Equivalently, g=f−1g = f^{-1} and f=g−1f = g^{-1}

  • Finding an inverse from a function written as y = f(x)

    • Step 1: Start with y=f(x)y = f(x)
    • Step 2: Swap x and y: x=f(y)x = f(y)
    • Step 3: Solve for y: y=f−1(x)y = f^{-1}(x)
    • The result f−1(x)f^{-1}(x) is the inverse function

  • Verifying inverses

    • Compose and check: f(f−1(x))=x ,f−1(f(x))=xf(f^{-1}(x)) = x\,,\quad f^{-1}(f(x)) = x
    • If either fails, the functions are not inverses over the stated domains

  • Transcript highlights (brief)

    • The instructor described solving for y to obtain the inverse expression
    • The exact algebra in the transcript was garbled (e.g., a line like "the inverse of x is equal to 1 minus 2 s all over x" is not reliable)

  • Homework logistics mentioned in transcript

    • Printing PDF homework to a paper copy for submission was suggested