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# Math - Operations on Functions, Kinds of Functions

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• ### Math - Operations on Functions, Kinds of Functions

1. 1. Graphs of Functions <ul><ul><li>Recall: </li></ul></ul><ul><ul><li>If f is a function from X to Y, then f = {(x,y) | y=f(x) and x in dom f } </li></ul></ul><ul><ul><li>(x,y) in R 2 means (x,y) is a point on the Cartesian plane </li></ul></ul><ul><ul><li>Def'n: </li></ul></ul><ul><ul><li>The graph of the function f is the set of points in the Cartesian plane having coordinates ordered pairs in f . </li></ul></ul>
2. 2. Graphs of Functions <ul><ul><li>Thus, we can find the range of a function f by graphing f . </li></ul></ul>
3. 3. Vertical Line Test <ul><ul><li>A vertical line intersects the graph of a function in at most one point </li></ul></ul><ul><ul><li>Examples: </li></ul></ul><ul><ul><li>Which of the following is a function: </li></ul></ul><ul><ul><li>1. f = { (x,y) | 3x + y – 2 = 0 } </li></ul></ul><ul><ul><li>2. f (x) = x 2 </li></ul></ul><ul><ul><li>3. {(x,y) | x 2 + y 2 = 1 } </li></ul></ul>
4. 4. Even and Odd Functions <ul><ul><li>Def'n: </li></ul></ul><ul><ul><li>Let f be a function. If for all x in dom f : </li></ul></ul><ul><ul><li>1. f (-x) = f (x) then f is an even function </li></ul></ul><ul><ul><li>2. f (-x) = - f (x) then f is an odd function </li></ul></ul><ul><ul><li>Determine whether even or odd: </li></ul></ul><ul><ul><li>1. f (x) = x 4 + 2x 2 + 3 </li></ul></ul><ul><ul><li>2. f (x) = x 3 + 3x </li></ul></ul>
5. 5. Odd and Even Functions <ul><ul><li>Thm: </li></ul></ul><ul><ul><li>Let f be a function. If f is </li></ul></ul><ul><ul><li>1. even then the graph of f is symmetric with respect to the y-axis </li></ul></ul><ul><ul><li>2. odd then the graph of f is </li></ul></ul><ul><ul><li>symmetric with respect to the origin </li></ul></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><li>1. f (x) = x 2 </li></ul></ul><ul><ul><li>2. f (x) = x 3 </li></ul></ul>
6. 6. Function Notation <ul><ul><li>Recall: y = f (x) if (x,y) is in f </li></ul></ul><ul><ul><li>Given f (x) = 2x 2 + 3 and g ((x) = 3x 1/2 . Find: </li></ul></ul><ul><ul><li>1. f (0) </li></ul></ul><ul><ul><li>2. g (-4) </li></ul></ul><ul><ul><li>3. f (c + 1) </li></ul></ul><ul><ul><li>4. f (c) + f (1) </li></ul></ul><ul><ul><li>5. f ( g (1)) </li></ul></ul>
7. 7. Operations on Functions <ul><ul><li>Let f and g be functions. Then: </li></ul></ul><ul><ul><li>1. ( f ∓ g ) (x) = f (x) ∓ g (x) </li></ul></ul><ul><ul><li> where dom( f ∓ g ) = dom f ∩ dom g </li></ul></ul><ul><ul><li>2. ( f g ) (x) = f (x) g (x) </li></ul></ul><ul><ul><li> where dom( f g ) = dom f ∩ dom g </li></ul></ul><ul><ul><li>3. ( f/g ) (x) = f (x) / g (x) </li></ul></ul><ul><ul><li> where dom( f/g ) = (dom f ∩ dom g ) {x | g (x) = 0} </li></ul></ul>
8. 8. Operations on Functions <ul><ul><li>4. ( f ∘ g ) (x) = f ( g (x)) </li></ul></ul><ul><ul><li>where dom( f ∘ g ) = {x | x in dom g and g (x) in dom f } </li></ul></ul><ul><ul><li>Given f (x) = 2x 2 + 3 and g ((x) = 3x 1/2 . </li></ul></ul><ul><ul><li>Find fg , f-g, f/g, f ∘ g </li></ul></ul>
9. 9. Constant Functions <ul><ul><li>functions of the form </li></ul></ul><ul><ul><li>f (x) = c </li></ul></ul><ul><ul><li>graph: horizontal line passing (0,c) </li></ul></ul><ul><ul><li>dom f = R </li></ul></ul><ul><ul><li>ran f = {c} </li></ul></ul><ul><ul><li>ie. f (x) = 2 </li></ul></ul>
10. 10. Linear Functions <ul><ul><li>functions of the form </li></ul></ul><ul><ul><li>f (x) = mx + b, m ≠ 0 </li></ul></ul><ul><ul><li>graph: line with slope m and y-intercept b </li></ul></ul><ul><ul><li>dom f = R </li></ul></ul><ul><ul><li>ran f = R </li></ul></ul><ul><ul><li>ie. f (x) = 2x + 3 </li></ul></ul>
11. 11. Quadratic Functions <ul><ul><li>functions of the form </li></ul></ul><ul><ul><li>f (x) = a x 2 + bx + c, a ≠ 0 </li></ul></ul><ul><ul><li>dom f = R </li></ul></ul><ul><ul><li>ran f = see the graph (later) </li></ul></ul><ul><ul><li>ie. f (x) = 2 x 2 + 3x + 4 </li></ul></ul>
12. 12. Graphing Quadratic Equations <ul><ul><li>Let f (x) = a x 2 + bx + c, a ≠ 0 </li></ul></ul><ul><ul><li>if </li></ul></ul><ul><ul><li>a < 0, parabola opens downward </li></ul></ul><ul><ul><li>a > 0, parabola opens upward </li></ul></ul><ul><ul><li>vertex of the parabola is at ( -b/2a, (4ac-b²)/4a) </li></ul></ul><ul><ul><li>if b²-4ac is </li></ul></ul><ul><ul><li>positive then the graph has two x-intercepts </li></ul></ul><ul><ul><li>zero then the vertex lies on the x-axis </li></ul></ul><ul><ul><li>negative then the graph has no x-intercepts </li></ul></ul>
13. 13. Absolute Value Functions <ul><ul><li>Recall: </li></ul></ul><ul><ul><li>f (x) = |x| = </li></ul></ul><ul><ul><li>x if x is positve </li></ul></ul><ul><ul><li>-x if x is negative </li></ul></ul><ul><ul><li>To graph absolute value functions, we graph the expression inside the absolute value sign first and make the negative y-values positive. Then we adjust... (see demo) </li></ul></ul>
14. 14. Piecewise Functions <ul><ul><li>functions defined differently for different intervals </li></ul></ul><ul><ul><li>ie. </li></ul></ul><ul><ul><li>f (x) = </li></ul></ul><ul><ul><li> 2x if x < -1 </li></ul></ul><ul><ul><li> x 2 if -1 ≤ x < 2 </li></ul></ul><ul><ul><li> -1 if x =2 </li></ul></ul><ul><ul><li> 2x-6 if x > 2 </li></ul></ul>