Determine the end behavior of the following monomial functions. (That is, does the function output increase without bound (→[infinity]) or decrease without bound (→−[infinity]) as the input increases/decreases without bound?)

Suppose f(x)=x2.

As x→[infinity], f(x)→
As x→−[infinity], f(x)→
Suppose g(x)=x3.

As x→[infinity], g(x)→
As x→−[infinity], g(x)→
Suppose h(x)=−6x3.

As x→[infinity], h(x)→
As x→−[infinity], h(x)→

a) f(x) = x²

As x→[infinity], f(x)→[infinity]

As x→−[infinity], f(x)→[infinity]

For this function, f(x) increases without bound as the input increases or decreases without bound. The graph of this function would be symmetric about the y-axis.

b) g(x) = x³

As x→[infinity], g(x)→[infinity]

As x→−[infinity], g(x)→-[infinity]

g(x) increases without bound as the input x increases without bound and decreases also without bound as input x decreases without bound. The graph of this function would be symmetric about the origin.

c) h(x)=−6x³.

As x→[infinity], h(x)→-[infinity]

As x→−[infinity], h(x)→[infinity]

h(x) decreases without bound as the input x increases without bound and increases without bound as input x decreases without bound. The graph of this function would also be symmetric about the origin.

Step-by-step explanation:

Normally, end behaviours predict the nature of the graphs of functions (especially as the values of x become very large, both in the positive and negative sense.

f(x) = x²

As x →[infinity],

f(x) = (∞)² → ∞

f(x) →[infinity]

And as x →−[infinity],

f(x) = (-∞)² → ∞

f(x) →[infinity]

For this function, f(x) increases without bound as the input increases or decreases without bound. The graph of this function would be symmetric about the y-axis.

b) g(x) = x³

As x→[infinity],

g(x) = (∞)³ → ∞

g(x)→[infinity]

As x→−[infinity],

g(x) = (-∞)³ → -∞

g(x)→−[infinity]

g(x) increases without bound as the input x increases without bound and decreases also without bound as input x decreases without bound. The graph of this function would be symmetric about the origin.

c) h(x)=−6x³.

As x→[infinity],

h(x) = -6(∞)³ → -6(∞) → -∞

h(x)→-infinity]

As x→−[infinity],

h(x) = -6(-∞)³ → -6(-∞) → ∞

h(x)→[infinity]

h(x) decreases without bound as the input x increases without bound and increases without bound as input x decreases without bound. The graph of this function would also be symmetric about the origin.

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step-by-step explanation: