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Algebra

Functions

Inverse Functions

         

Consider 3 functions f(x),g(x), f(x), g(x), and h(x) h(x) above.

Which has an inverse function ?

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Let f(x)=2x−7,f(x)=2x-7, and let g(x)=f−1(x)g(x)=f^{-1}(x). What is the value of g(15)+g−1(17)g(15)+g^{-1}(17)?

Details and assumptions

  • g−1(x)g^{-1}(x) denotes the inverse function of g(x)g(x).
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If f−1(2)=0f^{-1}(2)=0 and (f∘f)(0)=20(f \circ f)(0) = 20 for f(x)=ax+bf(x)=ax+b, what is a+ba+b?

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Consider the function f(x)= f(x) = {2x(x≥1)2(1−k)x+2k(x<1). \begin{cases} 2x & (x \ge 1) \\ 2(1-k)x + 2k & (x < 1). \end{cases}

If f−1(x) f^{-1}(x) exists, find the value of the constant k. k.

Details and assumptions

  • f−1(x) f^{-1}(x) denotes the inverse function of f(x). f(x).
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Consider a function f:X→Y, f: X \to Y, where X={x∣ 0≤x≤1} X = \{x |\, 0 \le x \le 1\} and Y={y∣ a≤y≤b}. Y = \{y|\, a \le y \le b\}.

If f(x)=3x+2, f(x) = 3x + 2, and f−1(x) f^{-1}(x) exists, find the values of a,b. a , b.

Details and assumptions

  • f−1(x) f^{-1}(x) denotes the inverse function of f(x). f(x).
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