3.4 Inverse Functions and Relations

Symmetry and Coordinate GraphsSection 3.1Symmetry with Respect to the OriginSymmetric with the origin if and only if the following statement is true:F(-x)=-F(x)Symmetric with Origin ExampleF(X) = x5YesF(x) = x/(1-x)NoSymmetry (a,b)X-AxisPlug in (a,-b)Y-AxisPlug in (-a,b)Y=XPlug in (b,a)Y=-XPlug in (-b,-a)ExampleDetermine whether the graph of xy=-2 is symmetric with respect to the x axis, yaxis, the line y=x, and the line y=-x, or none of these?First plug in (a,b)Ab=-2Symmetric with both line y=x and line y=-x

ExampleDetermine whether the graph of y =x+1 is symmetric with respect to the x axis, yaxis, both or neither?Symmetric with both the x and y axis.Even and Odd FunctionsEven Symmetric with respect to Y axisF(-x)=F(x)Odd Symmetric with respect to the originF(-x)=-F(x)F(X) = x5OddF(x) = x/(1-x)Neither odd nor evenEven and Odd FunctionsWhich lines are lines of symmetry for the graph of x2=1/y2X and y axises, y=x, and y=-xIs the following function symmetric about the origin?F(X)=-7x5 + 8xYes, does this mean its even or odd?OddFamilies of GraphsSection 3.2Parent Graphs Constant

Parent Graphs

ExampleGraph f(x) = x2 and g(x) = -x2. Describe how the graphs of f(x) and g(x) and are related.xf(x) = x2g(x) = -x2-24-4-11-100011-124-4

Changes to Parent Graph Graph Parent Graph of f(x)=|x|Graph f(x)=|x|+1Graph f(x) = |x|-1 Graph f(x)=|x+1|Graph f(x) = |x-1|On same graphSimilarities/Differences?Change to Parent GraphReflectionsY=-f(x)Outside the HVertical AxisReflected over the x-axisY=f(-x)Inside the HHorizontal AxisReflected over the y-axis

Change to Parent GraphTranslations+,- OUTSIDE of FunctionOutside the H Vertical MovementSHIFTS UP AND DOWN+,- INSIDE of FunctionInside the H Horizontal MovementSHIFTS LEFT AND RIGHTChange to Parent GraphDilationsX/ OUTSIDE of FunctionOutside the H Vertical MovementExpands/CompressesX/ INSIDE of FunctionInside the H Horizontal MovementExpands/Compresses

Examples - Use the parent graph y = x2 to sketch the graph of each function.

y = x2 + 1This function is of the form y = f(x) + 1. Outside the HVertical MovementSince 1 is added to the parent function y = x2,the graph of the parent function moves up 1 unit.

a.

Examples - Use the parent graph y = x2 to sketch the graph of each function.

y = (x - 2)2Inside the HHorizontal MovementThis function is of the form y = f(x - 2).Since 2 is being subtracted from x before being evaluated by the parent function,the graph of the parent function y = x2 slides 2 units right.

a.

Examples - Use the parent graph y = x2 to sketch the graph of each function.

y = (x + 1)2 2This function is of the form y = f(x + 1)2 -2.The addition of 1 indicates a slide 1 unit left, and the subtraction of 2 moves theparent function y = x2 down two units.

a.

EXAMPLESmake table and graph

Graphs of Nonlinear InequalitiesSection 3.3Determine which are solutionsDetermine whether (3, 4), (11, 2), (6, 5) and (18, -1) are solutions for the inequality y x-2) + 3.

Of these ordered pairs, (3, 4) and (6, 5) and are solutions for y x-2) + 3. ExampleDetermine whether (-2,5) (3,-1) (-4,2) and (-1,-1) are solutions for the inequality y 2x3+7(-2,5) and (-4,2) are solutionsGraph y (x - 2)2 + 2.

To verify numerically, you can test a point notin the boundary. It is common to test (0, 0) whenever it is not on the boundary

Since the boundary is included in the inequality,the graph is drawn as a solid curve.

Graph y < -2 - |x - 1|.

y < -2 - |x - 1|y < -|x - 1| - 2

The boundary is notincluded, so draw it as a dashed line

Verify by substituting (0, 0) in the inequality to obtain 0 < -3. Since this statement is false, the part of the graph containing (0, 0) should not be shaded. Thus, the graphis correct.

Solving Absolute InequalitiesSolve |x + 3| - 4 < 2.There are two cases that must be solved. In one case, x + 3 is negative, and in the other, x + 3 is positive.Case 1(x + 3)< 0|x + 3| - 4< 2-(x + 3) - 4< 2|x + 3| = -(x + 3)-x - 3 - 4< 2-x< 9x> -9Case 2(x + 3)> 0|x + 3| - 4< 2x + 3 - 4< 2|x + 3| = (x + 3)x - 1< 2x< 3The solution set is {x | -9 < x < 3}. {x | -9 < x < 3} is read as the set of all numbers x such that x is between -9 and 3.Solving Absolute InequalitiesSolve |x -2| - 5 < 4. -(x-2)-5