Question

Consider the following rational function fff. f(x)=\dfrac{6x^3-x^2+7}{2x+5}f(x)= 2x+5 6x 3 −x 2 +7 ​ f, left parenthesis, x, right parenthesis, equals, start fraction, 6, x, cubed, minus, x, squared, plus, 7, divided by, 2, x, plus, 5, end fraction Determine fff's end behavior. f(x)\tof(x)→f, left parenthesis, x, right parenthesis, \to as x\to -\inftyx→−∞x, \to, minus, infinity. f(x)\tof(x)→f, left parenthesis, x, right parenthesis, \to as x\to \inftyx→∞x, \to, infinity.

Answer 1

The end behavior of the rational polynomial function
`f(x) = (6x^3 - x^2 + 7)/(2x + 5)` is,

{x → ∞, y → ∞ and x → - ∞, y → ∞}.

What is the end behavior of a polynomial?

A polynomial function's final behavior is how its graph behaves as x gets closer to positive or negative infinity.

The graph's final behavior is determined by a polynomial function's degree and leading coefficient.

Given, A rational polynomial function
`f(x) = (6x^3 - x^2 + 7)/(2x + 5)`.

Now A cubic function divided by a linear function would result in a quadratic function, And as the coefficients of the highest degree terms of both the highest terms are positive the coefficient of the highest term of the quadratic function will also be positive and it's graph will be a parabola that opens upwards and symmetric about the y-axis.

Therefore, The end behavior will be when x tends to positive infinity y goes to positive infinity and when x tends to negative infinity y goes to positive infinity.

learn more about end behavior of polynomials here :