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Transcript
Functions and Graphs 1.2 Symmetric about the y axis FUNCTIONS Symmetric about the origin Even functions have y-axis Symmetry 8 7 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 So for an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph. Odd functions have origin Symmetry 8 7 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph. x-axis Symmetry We wouldn’t talk about a function with x-axis symmetry because it wouldn’t BE a function. 8 7 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 A function is even if f( -x) = f(x) for every number x in the domain. So if you plug a –x into the function and you get the original function back again it is even. f x 5 x 2 x 1 4 2 Is this function even? YES f x 5( x) 2( x) 1 5x 2 x 1 4 2 4 2 f x 2 x x Is this function even? NO 3 3 f x 2( x) ( x) 2 x x 3 A function is odd if f( -x) = - f(x) for every number x in the domain. So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd. f x 5 x 2 x 1 4 2 Is this function odd? NO f x 5( x) 2( x) 1 5x 2 x 1 4 2 4 2 f x 2 x x Is this function odd? YES 3 3 f x 2( x) ( x) 2 x x 3 If a function is not even or odd we just say neither (meaning neither even nor odd) Determine if the following functions are even, odd or neither. Not the original and all 3 terms didn’t change signs, so NEITHER. f x 5 x 1 f x 5 x 1 5 x 1 3 3 f x 3x x 2 4 2 Got f(x) back so EVEN. f x 3( x) ( x) 2 3x x 2 4 2 4 2 You should be familiar with the shapes of these basic functions. Library of Functions Equations that can be written f(x) = mx + b slope y-intercept The domain of these functions is all real numbers. f(x) = 3 f(x) = -1 f(x) = 1 Constant Functions f(x) = b, where b is a real number The domain of these functions is all real numbers. Would constant functions be even or odd or neither? The range will only be b If you put any real number in this function, you get the same real number “back”. f(x) = x Identity Function f(x) = x, slope 1, y-intercept = 0 The domain of this function is all real numbers. Would the identity function be even or odd or neither? The range is also all real numbers Square Function f(x) = x2 The domain of this function is all real numbers. Would the square function be even or odd or neither? The range is all NON-NEGATIVE real numbers Cube Function f(x) = x3 The domain of this function is all real numbers. The range is all real numbers Would the cube function be even or odd or neither? Square Root Function f x x The domain of this function is NON-NEGATIVE real numbers. Would the square root function be even or odd or neither? The range is NON-NEGATIVE real numbers 1 f x x Reciprocal Function The domain of this function is all NON-ZERO real numbers. Would the reciprocal function be even or odd or neither? The range is all NON-ZERO real numbers. f x x Absolute Value Function The domain of this function is all real numbers. Would the absolute value function be even or odd or neither? The range is all NON-NEGATIVE real numbers Recall: These are functions that are defined differently on different parts of the domain. x, x 0 f x 2 x , x 0 This means for x’s less than 0, put them in f(x) = -x but for x’s greater than or equal to 0, put them in f(x) = x2 What Whatdoes doesthe thegraph graph of off(x) f(x)==-x x2look looklike? like? Remember yy==f(x) f(x)so solets let’s Remember graphyy==x-2xwhich whichisisaasquare line of graph slope –1(parabola) and y-intercept 0. function Since Since we we are areonly only supposed supposed to tograph graphthis thisfor for xx< 0, we’ll only stop keep the graph the right at x =half 0. of the graph. This then is the graph for the piecewise function given above. 2 x 5, 3 x 0 f x 3, x0 5 x, x0 For Forxx>=00the thefunction function For x values between isvalue to be –3supposed andis0supposed graph the to along be so2x line plot ythe = - 5x. line–3 ythe = + 5. point (0, -3) Since you know this the graph graph is is aa piece of a line, you can just plug in each 0 to see endwhere value to to start get the the line and endpoints. then count af(-3) – 5 =slope. -1 and f(0) = 5 So this the graph of the piecewise function solid dot for "or equal to" open dot since not "or equal to" You try one: Graph the function described by: 3x x 3 h( x ) 2 x x 3 f g ( x) f ( g ( x)) “f of g of x” f ( x) x 7 g ( x) x 2 2 f ( g ( x)) x 7 f (g (2)) 3