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Graphical Solution of Equations with Domain and Range

Learn how to solve equations graphically by converting them into equivalent equations and finding the x-intercepts. Understand functions, their properties, domain, range, continuity, and discontinuity. Practice finding the domain and range of functions.

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Graphical Solution of Equations with Domain and Range

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  1. Do Now from 1.1b… Solve the equation graphically by converting it into an equivalent equation with 0 on the right-hand side and then finding the x-intercepts

  2. Functions and their properties…domain, range, continuity, discontinuity. We are functioning well in Sec. 1.2a!!!

  3. Definition: Function, Domain, and Range A function from a set D to a set R is a rule that assigns to every element in D a unique element in R. The set D of all input values is the domain of the function, and the set R of output values is therange of the function. Common notation: y = f(x) Here, x is the independent variable, and y is the dependent variable

  4. VLT!!! (not a tasty BLT sandwich…) Vertical Line Test: A graph (set of points (x, y)) in the x-y plane defines y as a function of x if and only if no vertical line intersects the graph in more than one point.

  5. Which of the following are graphs of functions? Yup!!! Nope!!! Heck Naw!!! Yessir!!!

  6. Finding Domain and Range Agreement for Domain: Unless we are dealing with a model (like volume) that necessitates a restricted domain, we will assume that the domain of a function defined by an algebraic expression is the same as the domain of the algebraic expression, the implied domain. For models, we will use a domain that fits the situation, the relevant domain.

  7. Finding Domain and Range Find the domain of the following functions (support graphically): The key question: Is there anything that x could not be??? Always write your answer in interval notation:

  8. Finding Domain and Range Find the domain of the following functions (support graphically): What are the restrictions on x ? Interval notation:

  9. Finding Domain and Range Find the range of the given function (use any method). What are the possible y-values for this function???

  10. Finding Domain and Range Find the range of the given function (use any method). Check the graph…

  11. The Concept of Continuity Algebraically, a function is continuous at x = a if Read “the limit of f (x) as x approaches a is f (a)” Graphically, a function is continuous at a particular point if the graph does not “come apart” at that point.  Let’s apply this with some examples…

  12. The Concept of Continuity Continuous at all x How does that “limit definition” apply???

  13. The Concept of Continuity Removable Discontinuity at x = a Why is it called “removable”? a

  14. The Concept of Continuity Jump Discontinuity at x = a a

  15. The Concept of Continuity Infinite Discontinuity at x = a a

  16. Homework: p. 98 1-23 odd

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