Question

in exercises 19–22, describe the end behavior of the function. example 3

1. h(x) = -5x⁴ + 7x³ - 6x² + 9x + 2
2. g(x) = 7x⁷ + 12x⁵ - 6x³ - 2x - 18
3. f(x) = -2x⁴ + 12x⁸ + 17 + 15x²
4. f(x) = 11 - 18x² - 5x⁵ - 12x⁴ - 2x

Explanation:

Step1: Identify leading term of $h(x)$

Leading term: $-5x^4$

Step2: Analyze degree and coefficient

Degree (4) even, coefficient (-5) negative.
As $x\to+\infty$, $h(x)\to-\infty$; as $x\to-\infty$, $h(x)\to-\infty$.

Step1: Identify leading term of $g(x)$

Leading term: $7x^7$

Step2: Analyze degree and coefficient

Degree (7) odd, coefficient (7) positive.
As $x\to+\infty$, $g(x)\to+\infty$; as $x\to-\infty$, $g(x)\to-\infty$.

Step1: Identify leading term of $f(x)$

Rearrange: $f(x)=12x^8-2x^4+15x^2+17$, leading term: $12x^8$

Step2: Analyze degree and coefficient

Degree (8) even, coefficient (12) positive.
As $x\to+\infty$, $f(x)\to+\infty$; as $x\to-\infty$, $f(x)\to+\infty$.

Step1: Identify leading term of $f(x)$

Rearrange: $f(x)=-5x^5-12x^4-18x^2-2x+11$, leading term: $-5x^5$

Step2: Analyze degree and coefficient

Degree (5) odd, coefficient (-5) negative.
As $x\to+\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to+\infty$.

Answer:

1. As $x\to+\infty$, $h(x)\to-\infty$; as $x\to-\infty$, $h(x)\to-\infty$
2. As $x\to+\infty$, $g(x)\to+\infty$; as $x\to-\infty$, $g(x)\to-\infty$
3. As $x\to+\infty$, $f(x)\to+\infty$; as $x\to-\infty$, $f(x)\to+\infty$
4. As $x\to+\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to+\infty$