Functions
Consider 3 functions \( f(x), g(x),\) and \( h(x) \) above.
Which has an inverse function ?
Let \(f(x)=2x-7,\) and let \(g(x)=f^{-1}(x)\). What is the value of \(g(15)+g^{-1}(17)\)?
\[\] Details and assumptions
- \(g^{-1}(x)\) denotes the inverse function of \(g(x)\).
If \(f^{-1}(2)=0\) and \((f \circ f)(0) = 20\) for \(f(x)=ax+b\), what is \(a+b\)?
Consider the function \( f(x) = \) \( \begin{cases} 2x & (x \ge 1) \\ 2(1-k)x + 2k & (x < 1). \end{cases} \)
If \( f^{-1}(x) \) exists, find the value of the constant \( k. \)
\[\] Details and assumptions
- \( f^{-1}(x) \) denotes the inverse function of \( f(x). \)
Consider a function \( f: X \to Y, \) where \( X = \{x |\, 0 \le x \le 1\} \) and \( Y = \{y|\, a \le y \le b\}. \)
If \( f(x) = 3x + 2, \) and \( f^{-1}(x) \) exists, find the values of \( a , b. \)
\[\] Details and assumptions
- \( f^{-1}(x) \) denotes the inverse function of \( f(x). \)