Respuesta :
Answer:
The value of b is -13
Step-by-step explanation:
* Lets explain how to find a and b
- In any rational function f(x), if the degree of the denominator =
the degree of the numerator, then there is a horizontal asymptote
at y = coefficient of higher x of the numerator ÷ coefficient of higher
x of the denominator
- Removable discontinuity means the numerator and denominator of
the rational function have common factor will reduce when we
simplify the fraction
- Discontinuity occurs when a number is both a zero of the numerator
and denominator
* Lets solve the problem
∵ [tex]f(x)=\frac{ax^{2}+bx+2}{2x^{2}-8}[/tex]
∵ The degree of numerator and denominator is 2
∵ The coefficient of x² in the numerator is a
∵ The coefficient of x² in the denominator is 2
∴ The horizontal asymptote is at [tex]y=\frac{a}{2}[/tex]
∵ The graph of f has a horizontal asymptote at y = 3
∴ [tex]\frac{a}{2}=3[/tex]
- By using cross multiplication
∴ a = 6
∴ [tex]f(x)=\frac{6x^{2}+bx+2}{2x^{2}-8}[/tex]
∵ The function f has a removable discontinuity at x = 2
∵ Discontinuity occurs when a number is both a zero of the numerator
and denominator
∴ 2 is a zero of the numerator
∴ 6x² + bx + 2 = 0 at x = 2
- Substitute x by 2 to find b
∴ 6(2)² + b(2) + 2 = 0
∴ 6(4) + 2b + 2 = 0
∴ 24 + 2b + 2 = 0
∴ 26 + 2b = 0
- Subtract 26 from both sides
∴ 2b = -26
- Divide both sides by 2
∴ b = -13
b = -13
Step-by-step explanation:
Given :
[tex]\rm f(x) = \dfrac{ax^2+bx+2}{2x^2-8}[/tex] ----- (1)
The graph of f has a horizontal asymptote at y = 3 and f has a removable discontinuity at x = 2.
Solution :
Horizontal assymptote is
[tex]y = \dfrac{a}{2}[/tex]
because in equation (1) coefficient of [tex]x^2[/tex] is 'a' is in numerator and '2' is in denomenator.
So,
a = 6
Now equation (1) becomes,
[tex]\rm f(x) = \dfrac{6x^2+bx+2}{2x^2-8}[/tex] ------ (2)
Given that discontinuity at x = 2, so denomenator of equation (2) is zero at x = 2. Therefore
[tex]\rm 6x^2 + bx + 2 = 0[/tex]
[tex]6(2^2) + 2b +2 = 0[/tex]
-26 = 2b
b = -13
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https://brainly.com/question/22421166?referrer=searchResults
