Summary

The domain of a function is the value of \(x\) for which the function is defined. For example, \(f \left(x\right) = x^{2} + 1\) for \(x \geq 0\) (domain is \(x \geq 0\)) \(g \left(x\right) = \frac {1}{x-1} + \frac {1}{x-2} \) for \(1 \lt x \lt 2\) (domain is \(1 \lt x \lt 2\))

The range of a function is the set of values that the function maps onto.
For example, if \(f \left(x\right) = x^{2}\), \(0 \leq x \leq 5\),
the range of \(f\) is \(0 \leq f \left(x\right) \leq 25\).

1 : 1 mapping is a function for which every value of \(f\left(x\right)\) is unique; that is, if \(f \left(a\right) \neq f \left(b\right) \) unless \(a = b\)
For example, \(f \left(f\right) = x + 1\) is a 1 : 1 mapping, but
\(f \left(x\right) = x^{2}\)is not a 1 : 1 mapping as, for example, \(f \left(2\right) = 4 = f\left(-2\right)\)

The composite function \(f g\) or \(f g\left(x\right)\) is defined as \(f \left(g\left(x\right)\right)\)
For example, if \(f \left(x\right) = x +1\) and \(g \left(x\right) = x^{2}\), then \(f \left(x\right) = f \left(x^{2}\right) = x^{2} + 1\)
and \(gf \left(x\right) = g \left(x +1\right) = \left(x + 1\right) ^{2}\)

The inverse function of \(f\) is denoted by \(f^{-1}\) and is such that \( f f^{-1} \left(x\right) = x = f^{-1} f \left(x\right)\)
The inverse only exists if the function is a 1 : 1 mapping.

The graph of \(y = f\left(x\right)\) is mapped onto the graph of \(y= f\left(x\right) + 2\) by translating it up \(2\;units\)
In general \(f \left(x\right) + a\) moves a curve up \(a\;units\) and \(f\left(x\right) -a\) moves it down \(a\;units\), where \(a\) is a positive number.

The graph of \(y = f \left(x\right)\) is mapped onto \(f\left(x+1\right)\) by a translation of \(1\;unit\) to the left.

In general \(f \left(x+a\right)\) translates a curve \(a\) units to the left and \(f \left(x-a\right)\) translates a curve units to the right, where a is a positive number.

The curve for \(f\left(2x\right)\) is much steeper than for \(f\left(x\right)\). This is because the curve has been compressed by a factor of \(2\) in the \(x-direction\). Compare the rectangles \(ABCD\) and \(EFGH\).

In general the curve of \(y = f \left(kx\right)\) will be compressed by a factor of \(k\) in the \(x-direction\), where \(k \gt 1\).

Here the curve \(y = f \left(x\right)\) has been stretched by a factor of \(2\) in the vertical or \(y-direction\) to obtain the curve \(y = f \left(x\right)\). Compare the rectangles \(ABCD\) and \(CDFE\).

In general the curve of \(y = kf \left(x\right)\) stretches the graph of \(y = f \left(x\right)\) by a factor of \(k\) in the \(y-direction\) where \(k \gt 1\).

Note that if \(k\) is negative and \(k \lt −1\), the curve will be stretched and reflected in the \(x-axis\), while if \(−1 \lt k \lt 1\), it is compressed.

Functions

Welcome