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Essential Concepts
Introduction to Functions
- A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.
- Function notation is a shorthand method for relating the input to the output in the form [latex]y=f\left(x\right)[/latex].
- In table form, a function can be represented by rows or columns that relate to input and output values.
- To evaluate a function we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.
- To solve for a specific function value, we determine the input values that yield the specific output value.
- An algebraic form of a function can be written from an equation.
- Input and output values of a function can be identified from a table.
- Relating input values to output values on a graph is another way to evaluate a function.
- A function is one-to-one if each output value corresponds to only one input value.
- A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.
- A graph represents a one-to-one function if any horizontal line drawn on the graph intersects the graph at no more than one point.
- A function may be described using notation stating that the output [latex]y[/latex] is dependent upon the input [latex]x[/latex] by writing [latex]y = f(x)[/latex].
- The vertical line test involves drawing vertical lines through the graph. If any vertical line intersects the graph at more than one point, the graph does not represent a function. If every vertical line intersects the graph at most one point, the graph represents a function.
- When working with functions, it is similarly helpful to have a base set of building- block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Toolkit functions are the constant function, linear (identity) function, absolute value function, quadratic function, cubic function, reciprocal function, reciprocal-squared function, square root function, and cube root function.
Domain and Range
- The set of all input values makes up the domain, and the set of all output variables makes up the range of the function.
- The domain of a function can be determined by listing the input values of a set of ordered pairs.
- The domain of a function can also be determined by identifying the input values of a function written as an equation.
- Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation.
- For many functions, the domain and range can be determined from a graph.
- An understanding of toolkit functions can be used to find the domain and range of related functions.
- A piecewise function is described by more than one formula.
- A piecewise function can be graphed using each algebraic formula on its assigned subdomain.
Rates of Change and Behavior of Graphs
- A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data.
- Identifying points that mark the interval on a graph can be used to find the average rate of change.
- Comparing pairs of input and output values in a table can also be used to find the average rate of change.
- An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula.
- The average rate of change can sometimes be determined as an expression.
- We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. Lastly, a function is constant on an interval if the output values are the same for all input values so the slope is zero. It is neither increasing nor decreasing like the graph of a constant function.
- A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.
- A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.
- Minima and maxima are also called extrema.
- We can find local extrema from a graph.
- To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval.
Key Equations
| Average rate of change | [latex]\frac{\Delta y}{\Delta x}=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}[/latex] |
| Constant function | [latex]f\left(x\right)=c[/latex], where [latex]c[/latex] is a constant |
| Identity function | [latex]f\left(x\right)=x[/latex] |
| Absolute value function | [latex]f\left(x\right)=|x|[/latex] |
| Quadratic function | [latex]f\left(x\right)={x}^{2}[/latex] |
| Cubic function | [latex]f\left(x\right)={x}^{3}[/latex] |
| Reciprocal function | [latex]f\left(x\right)=\frac{1}{x}[/latex] |
| Reciprocal squared function | [latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex] |
| Square root function | [latex]f\left(x\right)=\sqrt{x}[/latex] |
| Cube root function | [latex]f\left(x\right)=\sqrt[3]{x}[/latex] |
Glossary
- absolute maximum
- the greatest value of a function over an interval
- absolute minimum
- the lowest value of a function over an interval
- average rate of change
- the difference in the output values of a function found for two values of the input divided by the difference between the inputs
- decreasing function
- a function is decreasing in some open interval if [latex]f\left(b\right)
dependent variable
an output variable
- domain
- the set of all possible input values for a relation
- function
- a relation in which each input value yields a unique output value
- horizontal line test
- a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once
- increasing function
- a function is increasing in some open interval if [latex]f\left(b\right)>f\left(a\right)[/latex] for any two input values [latex]a[/latex] and [latex]b[/latex] in the given interval where [latex]b>a[/latex]
- independent variable
- an input variable
- input
- each object or value in a domain that relates to another object or value by a relationship known as a function
- interval notation
- a method of describing a set that includes all numbers between a lower limit and an upper limit; the lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion
- local extrema
- collectively, all of a function’s local maxima and minima
- local maximum
- a value of the input where a function changes from increasing to decreasing as the input value increases.
- local minimum
- a value of the input where a function changes from decreasing to increasing as the input value increases.
- one-to-one function
- a function for which each value of the output is associated with a unique input value
- output
- each object or value in the range that is produced when an input value is entered into a function
- piecewise function
- a function in which more than one formula is used to define the output
- range
- the set of output values that result from the input values in a relation
- rate of change
- the change of an output quantity relative to the change of the input quantity
- relation
- a set of ordered pairs
- set-builder notation
- a method of describing a set by a rule that all of its members obey; it takes the form [latex]\left\{x|\text{statement about }x\right\}[/latex]
- vertical line test
- a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once