Summary and Main Ideas

• Function: a rule which uniquely associates elements of one set \(A\) with the elements of another set \(B\); each element in set \(A\) maps to only one element in set \(B\).

• Functions can be one-to-one relations or many-to-one relations. A many-to-one relation associates two or more values of the independent (input) variable with a single value of the dependent (output) variable.

• Vertical line test: if it is possible to draw any vertical line which crosses the graph of the relation more than once, then the relation is not a function.

• Given the invertible function \(f(x)\), we determine the inverse \(f^{-1}(x)\) by:

  • replacing every \(x\) with \(y\) and \(y\) with \(x\);
  • making \(y\) the subject of the equation;
  • expressing the new equation in function notation.

• If we represent the function \(f\) and the inverse function \({f}^{-1}\) graphically, the two graphs are reflected about the line \(y=x\).

• The domain of the function is equal to the range of the inverse. The range of the function is equal to the domain of the inverse.

• The inverse function of a straight line is also a straight line. Vertical and horizontal lines are exceptions.

• The inverse of a parabola is not a function. However, we can limit the domain of the parabola so that the inverse of the parabola is a function.

• The inverse of the exponential function \(f(x) = b^{x}, (b > 0, b \ne 1)\) is the logarithmic function \(f^{-1}(x) = \log_{b}{x}\).

• The “common logarithm” has a base \(\text{10}\) and can be written as \(\log_{10}{x} = \log{x}\). The \(\log\) symbol written without a base means \(\log\) base\(\text{10}\).

Logarithmic Laws:

• \(\log_{a}{x^{b}} = b \log_{a}{x} \qquad (x > 0)\)
• \(\log_{a}{x} = \cfrac{\log_{b}{x}}{\log_{b}{a}} \qquad (b > 0 \text{ and } b \ne 1)\)
• \(\log_{a}{xy} = \log_{a}{x} + \log_{a}{y} \qquad (x > 0 \text{ and } y > 0)\)
• \(\log_{a}{\cfrac{x}{y}} = \log_{a}{x} - \log_{a}{y} \qquad (x > 0 \text{ and } y > 0)\)

	Straight line function	Quadratic function	Exponential function
Formula	\(y = ax + q\)	\(y = ax^{2}\)	\(y = b^{x}\)
Inverse	\(y = \cfrac{x}{a} - \cfrac{q}{a}\)	\(y = \pm \sqrt{\cfrac{x}{a}}\)	\(y = \log_{b}{x}\)
Inverse a function?	yes	no	yes
Graphs