Section 3.6 Reciprocal Functions Section 3.6 Reciprocal Functions

Objectives: 1.To identify vertical asymptotes, domains, and ranges of reciprocal functions. 2.To graph reciprocal functions. Objectives: 1.To identify vertical asymptotes, domains, and ranges of reciprocal functions. 2.To graph reciprocal functions.

Reciprocal Function Any function that is a reciprocal of another function. DefinitionDefinition

Reciprocal trigonometric ratios: tan x 1 1 x x cot = = sin x 1 1 x x csc = = cos x 1 1 x x sec = = DefinitionDefinition

Reciprocal trigonometric functions: y = sec  y = csc  y = cot  Reciprocal trigonometric functions: y = sec  y = csc  y = cot  DefinitionDefinition

These functions are examples of a larger class of reciprocal functions, including reciprocals of power, polynomial, and exponential functions.

Examples of reciprocal functions g(x) = 1 x 2 – 4 1 x 2 – 4 f(x) = 1 3x 4 1 3x 4 h(x) = 14x14x 14x14x k(x) = sec x

EXAMPLE 1 Find f(1), g(2), h( 1 / 2 ), and k(  / 4 ), using the functions above. f(x) = 1 3x 4 1 3x 4 f(1) = = 1 3(1) 4 1 3(1)

EXAMPLE 1 Find f(1), g(2), h( 1 / 2 ), and k(  / 4 ), using the functions above. g(x) = 1 x 2 – 4 1 x 2 – 4 g(2) = =, which is undefined – –

EXAMPLE 1 Find f(1), g(2), h( 1 / 2 ), and k(  / 4 ), using the functions above. h(x) = 14x14x 14x14x h( 1 / 2 ) = = = 1 4 1/ /

EXAMPLE 1 Find f(1), g(2), h( 1 / 2 ), and k(  / 4 ), using the functions above. k(x) = sec x k(  / 4 ) = sec  / 4 = = 1 cos  / 4 1 cos  / / / =

Since reciprocal functions have denominators, you must be careful about what values are used in the domain.

EXAMPLE 2 Find the domains of f(x), g(x), h(x), and k(x) in the previous functions. Find all values for which the denominator of f(x) and g(x) equals zero. f(x) 3x 4 = 0 x 4 = 0 x = 0 f(x) 3x 4 = 0 x 4 = 0 x = 0 g(x) x 2 – 4 = 0 x 2 = 4 x = ±2 g(x) x 2 – 4 = 0 x 2 = 4 x = ±2

EXAMPLE 2 Find the domains of f(x), g(x), h(x), and k(x) in the previous functions. Exclude those values from the domain. f(x): D = {x|x  R, x ≠ 0} g(x): D = {x|x  R, x ≠ ±2}

EXAMPLE 2 Find the domains of f(x), g(x), h(x), and k(x) in the previous functions. Since 4 x ≠ 0  x, the domain of h(x) is R. Since cos x = 0 when x =  /2 + k , k  R, the domain of k(x) is D = {x|x  R, x ≠  /2 + k , k  Z }. Since cos x = 0 when x =  /2 + k , k  R, the domain of k(x) is D = {x|x  R, x ≠  /2 + k , k  Z }.

EXAMPLE 3 Graph g(x) =. 1 x 2 – 4 1 x 2 – 4 Give the domain and range. Is g(x) continuous? Is g(x) an odd or even function?

Use reciprocal principles to graph g(x). EXAMPLE 3 Graph g(x) =. 1 x 2 – 4 1 x 2 – 4

Use reciprocal principles to graph g(x). EXAMPLE 3 Graph g(x) =. 1 x 2 – 4 1 x 2 – 4

D = {x|x ≠ ±2} R = {y|y  0 or y  - 1 / 4 } g(x) is an even function but is not continuous. D = {x|x ≠ ±2} R = {y|y  0 or y  - 1 / 4 } g(x) is an even function but is not continuous. EXAMPLE 3 Graph g(x) =. 1 x 2 – 4 1 x 2 – 4

again without graphing its reciprocal function first. 1.Find the domain excluding values where the denominator equals zero. x 2 – 4 = 0 x 2 = 4 x = ±2 D = {x|x ≠ ±2} 1.Find the domain excluding values where the denominator equals zero. x 2 – 4 = 0 x 2 = 4 x = ±2 D = {x|x ≠ ±2} EXAMPLE 4 Graph g(x) =. 1 x 2 – 4 1 x 2 – 4

again without graphing its reciprocal function first. 2.Check for x-intercepts. Since the numerator cannot equal zero, the graph cannot touch the x-axis. EXAMPLE 4 Graph g(x) =. 1 x 2 – 4 1 x 2 – 4

again without graphing its reciprocal function first. 3.Plot a point in each of the regions determined by the asymptotes (2 & -2). Since the graph cannot cross the x- axis, points within a region will all be on the same side of the x-axis. Include the y-intercept as one of the points. EXAMPLE 4 Graph g(x) =. 1 x 2 – 4 1 x 2 – 4

again without graphing its reciprocal function first. 4.Use the asymptotes as guides. Your graph will never quite reach either vertical asymptote or the x-axis. EXAMPLE 4 Graph g(x) =. 1 x 2 – 4 1 x 2 – 4

EXAMPLE 4 Graph g(x) =. 1 x 2 – 4 1 x 2 – 4

EXAMPLE 5 Graph y = csc x. Give the domain, range, zeros, and period. Is it continuous?

D = {x|x ≠ k , k  Z } R = {y|y  1 or y  -1} The function has no zeros; the period is 2 . It is not continuous. D = {x|x ≠ k , k  Z } R = {y|y  1 or y  -1} The function has no zeros; the period is 2 . It is not continuous.

Homework: pp Homework: pp

►A. Exercises 1.f(x) ►A. Exercises 1.f(x)

►A. Exercises 3.p(x) ►A. Exercises 3.p(x)

►A. Exercises 6. Give the vertical asymptotes of ►A. Exercises 6. Give the vertical asymptotes of h(x) = 1 x 2 + 5x – 14 1 x 2 + 5x – 14

►A. Exercises Evaluate each function as indicated. 9.f(x) = for x = 2 and x = -6 ►A. Exercises Evaluate each function as indicated. 9.f(x) = for x = 2 and x = -6 1 x 2 – 25 1 x 2 – 25

►B. Exercises 12. Graph the reciprocal function. Give the domain and range. ►B. Exercises 12. Graph the reciprocal function. Give the domain and range. h(x) = 1 x 2 + 5x – 14 1 x 2 + 5x – 14

■ Cumulative Review 41.Solve  ABC where A = 58°, B = 39°, and a = ■ Cumulative Review 41.Solve  ABC where A = 58°, B = 39°, and a = 10.5.

■ Cumulative Review 42.Give the period and amplitude of y = 5 sin 3 x. ■ Cumulative Review 42.Give the period and amplitude of y = 5 sin 3 x.

■ Cumulative Review 43.Find f (4) if ■ Cumulative Review 43.Find f (4) if f(x) =   x – 8if x  3 x 2 – 1if 3  x  9 7xif x  9 x – 8if x  3 x 2 – 1if 3  x  9 7xif x  9

■ Cumulative Review 44.How many zeros does a cubic polynomial function have? Why? ■ Cumulative Review 44.How many zeros does a cubic polynomial function have? Why?

■ Cumulative Review 45.Graph y = 2 x and estimate from the graph. ■ Cumulative Review 45.Graph y = 2 x and estimate from the graph.

A summary of principles for graphing reciprocal functions follows: 1.The larger the number, the closer the reciprocal is to zero. 2.The reciprocal of 1 and -1 is itself. 3.There is a vertical asymptote for the reciprocal when f(x) = 0. A summary of principles for graphing reciprocal functions follows: 1.The larger the number, the closer the reciprocal is to zero. 2.The reciprocal of 1 and -1 is itself. 3.There is a vertical asymptote for the reciprocal when f(x) = 0.