Inverse Functions (Leaving Cert Mathematics): Flashcards

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Inverse Functions
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Purpose of inverse function

Reverses input-output relationship

Mapping of f−1(x)f^{-1}(x) if f(x)f(x) maps x→yx \to y

y→xy \to x

Step 1 to find inverse

Replace f(x)f(x) with yy

Step 2 to find inverse

Swap xx and yy

Step 3 to find inverse

Solve for yy

Step 4 to find inverse

Replace yy with f−1(x)f^{-1}(x)

f−1(x)f^{-1}(x) if f(x)=2x+3f(x) = 2x + 3

x−32\frac{x-3}{2}

Requirement for function to have inverse

Must be one-to-one (1-to-1)

Inverse of f(x)=x2,x∈Rf(x) = x^2, x \in \mathbb{R}

No inverse exists

Why f(x)=x2f(x) = x^2 has no inverse

It is many-to-one

Geometric meaning of swapping xx and yy

Reflecting graph through line y=xy = x

Line of reflection for function and its inverse

y=xy = x

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