Inequalities of combined functions
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This document discusses techniques for illustrating inequalities between combined functions graphically and algebraically. It presents three graphical methods - comparing functions visually with different colors, analyzing the difference function by subtracting one from the other, and analyzing the quotient of the functions. The document also provides an example of identifying points of intersection between two functions, f(x) = x and g(x) = (x-2)2, graphically and algebraically to verify the solutions.
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Domain-and-Range-of-a-Function
The document discusses domain and range of functions. It provides examples of determining the domain and range from graphs of functions and from algebraic rules that define functions. The domain of a function is the set of permissible input values, while the range is the set of permissible output values. Examples show how to identify the domain and range from graphs by considering limits, and how to determine them algebraically by manipulating equations to solve for variables.
Appendex f
This document describes rectangular coordinate systems and how to plot points and graphs in them. It contains the following key points:
- A rectangular coordinate system uses two perpendicular axes (typically x and y) that intersect at the origin to locate points in a plane.
- The coordinates of a point P are ordered pairs (x,y) where x is the point's distance from the y-axis and y is its distance from the x-axis.
- An equation in x and y defines a graph - the set of all points whose coordinates satisfy the equation. Graphs can be approximated by plotting sample points but this has limitations.
- Intercepts are points where a graph crosses an axis,
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3 2 Polynomial Functions And Their Graphs
Polynomial functions have graphs that are smooth and continuous curves without sharp corners or breaks. Odd-degree polynomials have opposite behavior at their ends, while even-degree polynomials have the same behavior. The x-intercepts of a polynomial function are its zeros, which are found by setting the polynomial equal to 0. A polynomial function's graph can have at most n-1 turning points if the function is of degree n.
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This document defines key terms related to functions such as domain, range, and piecewise functions. It provides examples of representing functions using tables, ordered pairs, graphs, and equations. It also discusses how to determine if a relation represents a function and describes piecewise functions as using more than one formula with separate domains.
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Inequalities of combined functions
- 1. Inequalities of Combined Functions Example 1 TECHNIQUES FOR ILLUSTRATING INEQUALITIES Section 8.4
- 2. Graphing to Identify P.O.I’s Let f(x) = x and g(x) = (x-2)2
- 3. Graphing to Identify P.O.I From graphs we can see that P.O.I’s are (1,1) and (4,4) We can verify this algebraically:
- 4. Illustrating Graph Regions: Method 1 Compare the function visually Different colours represent different regions of graph Easy to visualize regions where one function is greater than the other
- 5. Analyze the Difference Function Subtract the functions and see where the graph of the difference is above the x-axis Illustrating Graph Regions: Method 2
- 7. Illustrating Graph Regions: Method 3 F(x) and g(x) can also be compared by analysing their quotient. Graph the combined function and identify the intervals for which This quotient is greater than one, which will correspond to where f(x) > g(x)
- 8. Conclusively, algebraic and graphical representations of inequalities can be useful for solving problems involving combined functions.