1) Find the domain and range of the rational function y( x) = x^2-25 / 2x^2...

1) Find the domain and range of the rational function y( x) = x^2-25 / 2x^2 + 13x+15

A) Factor the numerator and denominator

B) Determine the point of discontinuity if it exists.

Solutions

Expert Solution

Domain is the set of real values of x for which the function is real and defined

Since the factor (x + 5) is common in the numerator and denominator of y(x), this implies there is a hole at x = -5. Therefore, the function is not defined at x = -5

Also, for y(x) to be defined, the denominator cannot be equal to 0

Therefore,

Thus,

Domain is (- , -5) U (-5, -3 / 2) U (-3 / 2, )

Range is the set of real values the function can take in the defined domain.

Substituting x = -5 in y(x) we get

=> y(-5) = 10 / 7

Also, the horizontal asymptote of the function is given by

y = leading coefficeint of numerator / leading coefficient of denominator

=> y = 1 / 2

Therefore,

Range is (- , 1 / 2) U (1 / 2, 10 / 7) U (10 / 7, )

B) Since the factor (x + 5) is common in the numerator and denominator of y(x), this implies there is a discontinuity at x = -5

Point of discontinuity is (-5, 10 / 7)

milcah answered 1 year ago

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