Question

Classify each of the power functions based on their end behavior (increasing or decreasing) as x = ∞

Answer 1

Answer: right side behavior:

              f(x) is Decreasing

             g(x) is Increasing

             h(x) is Increasing

             j(x) is Decreasing

Step-by-step explanation:

The rules for end behavior are based on 2 criteria: Sign of leading coefficient and Degree of polynomial

Sign of leading coefficient (term with greatest exponent):

• If sign is positive, then right side is increasing
• If sign is negative, then right side is decreasing

Degree of polynomial (greatest exponent of polynomial:

• If even, then end behavior is the same from the left and right
• If odd, then end behavior is opposite from the left and right

f(x) = -2x²

• Sign is negative so right side is decreasing
• Degree is even so left side is the same as the right side (decreasing)

as x → +∞, f(x) → +∞  Decreasing

as x → -∞, f(x) → -∞   Decreasing

g(x) = (x + 2)³

• Sign is positive so right side is increasing
• Degree is odd so left side is opposite of the right side (decreasing)

as x → +∞, f(x) → +∞  Increasing

as x → -∞, f(x) → -∞   Decreasing

$h(x)=-1+x^{\frac{1}{2}}\implies h(x)=x^{\frac{1}{2}}-1$  

• Sign is positive so right side is increasing
• Degree is an even fraction so left side is opposite of the right side as it approaches the y-intercept (-1)

as x → +∞, f(x) → +∞  Increasing

as x → -∞, f(x) → -1    Decreasing to -1

$j(x)=\dfrac{1}{2}(-x)^5\implies j(x)=\dfrac{1}{2}(-1)^5(x)^5\implies j(x)=-\dfrac{1}{2}x^5$

• Sign is negative so right side is decreasing
• Degree is odd so left side is opposite of the right side (increasing)

as x → +∞, f(x) → +∞  Decreasing

as x → -∞, f(x) → -∞   Increasing