functions-domain-and-range.docx

Mathematics Learning Centre =========================== Functions: The domain and range Ⓧc 2006 University of Sydney ============================ 1. **Functions** 1. What is a function? ------------------- 1. ### Definition of a function *X Y X Y* ### The Vertical Line Test ![](med...

Mathematics Learning Centre =========================== Functions: The domain and range Ⓧc 2006 University of Sydney ============================ 1. **Functions** 1. What is a function? ------------------- 1. ### Definition of a function *X Y X Y* ### The Vertical Line Test  ### Domain of a function ### Range of a function ### Example a. State the domain and range of *y* = *√x* + 4. b. Sketch, showing significant features, the graph of *y* = *√x* + 4. ### Solution ### b. ### Example a. A parabola, which has vertex (3*, −*3), is sketched below. b. Find the domain and range of this function. ### Solution ### Example a. the domain and range b. *f* (*q*) c. *f* (*x*^2^). ### Solution The graph of *f* (*x*) = 3*x − x*^2^. a. The domain is all real *x*. The range is all real *y* where *y ≤* 2*.*25. b. *f* (*q*) = 3*q − q*^2^ ### Example  The graph of *f* (*x*) = (*x −* 1)^2^ + 1. ### Solution Specifying or restricting the domain of a function -------------------------------------------------- *y* = *x*^2^ for 0 *≤ x ≤* 2 The graph of *y* = *x*^2^ for 0 *≤ x ≤* 2. Exercises --------- 1. **a.** State the domain and range of *f* (*x*) = *√*9 *− x*2. b. Sketch the graph of *y* = *√*9 *− x*2. 2. Sketch the following functions stating the domain and range of each: a. *y* = *√x −* 1 b. *y* = *\|*2*x\|* c. *y* = ^1^ 3. Explain the meanings of function, domain and range. Discuss whether or not *y*^2^ = *x*^3^ 4. Sketch the following relations, showing all intercepts and features. State which ones are functions giving their domain and range. c. *y* = *x*^3^ d. *y* = [ *^x^*] *,x* = 0 e. *\|y\|* = *x*. 5. Write down the values of *x* which are not in the domain of the following functions: Solutions to exercises 1.3 -------------------------- 1. **a.** The domain of *f* (*x*) = *√*9 *− x*2 is all real *x* where *−*3 *≤ x ≤* 3. The range is all real *y* such that 0 *≤ y ≤* 3. ### b. ### a. b. ### c. ### d. 3. *y*^2^ = *x*^3^ is not a function. If *x* = 1, then *y*^2^ = 1 and *y* = 1 or *y* = *−*1. ### c. d. ### e.