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Revision:Algebraic Functions

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Algebraic Function

Functions

A function is a rule which indicates an operation to perform.

e.g. if f(x) = x^2 + 3 then f(2) = 2^2 + 3 = 7 (i.e. replace x with 2).

Functions can be graphed. For example, the graph of f(x) = 1/x is as follows:

Types of graphs

The graph of y = k/x is known as a hyperbola. An asymptote is a line (or a curve) that the curve gets arbitrarily close to; in the case of the hyperbola, the lines x=0 and y=0 are asymptotes.

Parabolas are graphs of the form y=ax^2+bx+c (where a, b and c are numbers, with $a \neq 0$ ). They can be $\cup$ shaped, when a is positive, or $\cap$ shaped, when a is negative.

Graph transformations

y = f(x) + a is the same as the graph y = f(x), shifted upwards by a units.

y = f(x - a) shifts the graph a units to the right.

y = f(ax) is a stretch with scale factor 1/a parallel to the x-axis.

y = a.f(x) is a stretch with scale factor a parallel to the y-axis.

Inverse Functions

The inverse function of a function f(x) is written $f^{-1}(x)$ . The inverse of a function does the opposite of the function - that is, $f(f^{-1}(x)) = f^{-1}(f(x)) = x$ . So, for example, if f(x) = 2x, we have $f^{-1}(x) = \frac{x}{2}$ .

To find the inverse of a function, follow the following procedure: let y =f(x). Swap all y's and x's . Rearrange to give y in terms of x. This is the inverse function.

Example:

Find the inverse of f(x), where f(x) = 3x - 7.

f(x) = 3x - 7.

y = 3x - 7

Swap x and y: x = 3y - 7

Rearrange to give y in terms of x: y = (x+7)/3

And so the inverse function is $f^{-1}(x) = \frac{x+7}{3}$ .