Determine the end behavior of each function as x →+∞and as x →-∞(a) f(x)=x^7 (b) f(x)=3 x+7

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Determine the end behavior of each function as x →+∞and as x...

Determine the end behavior of each function as $x \rightarrow$ $+\infty$ and as $x \rightarrow-\infty$
(a) $\quad f(x)=x^{7}$
(b) $\quad f(x)=3 x+7 x^{3}-12 x^{4}$
(c) $\quad f(x)=x^{-4}$
(d) $\quad f(x)=\frac{6 x^{3}-5 x^{2}+2}{x^{3}-8}$

Key Concepts

Rational Functions

Rational functions, which are ratios of polynomials, have end behavior that depends on the degrees of the numerator and denominator. When the degrees are equal, the function approaches the ratio of the leading coefficients; when the degree of the numerator is lower, the function typically approaches zero; and when the degree of the numerator is higher, the function exhibits unbounded behavior.

Leading Coefficient Test

The leading coefficient test is used to determine the end behavior of polynomial functions by examining the sign of the leading coefficient and the degree of the polynomial. The parity of the degree (odd or even) combined with the sign gives insight into how the function behaves as x approaches infinity or negative infinity.

Functions with Negative Exponents

Functions involving negative exponents, such as reciprocal power functions, typically tend to zero as the variable grows large in magnitude. This occurs because the denominator increases without bound, causing the overall value of the function to diminish towards zero.

End Behavior

End behavior refers to the trend of a function’s values as the independent variable approaches positive or negative infinity. It summarizes how the function behaves far from the origin, which is crucial in understanding the overall structure of the graph.

Dominant Term in Polynomials

For polynomial functions, the term with the highest degree (the dominant term) determines the end behavior. This is because as the magnitude of the variable increases, the highest degree term grows faster than the others and thus dictates the function’s overall trend.

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