Functions

In the realm of mathematics, functions serve as key concepts, which link each input with a single, unique output. They are typically denoted as 'f(x)', with 'x' representing the input and 'f(x)' illustrating the output. Functions are instrumental in explaining the correlations existing between variables in mathematical equations. They can be demonstrated visually through graphs or articulated through specific formulas.

Determining if the Point is on the Graph

Given the function

f (x) = x^{3} + 2 x^{2} - x + 2

, determine if the point

(2, 14)

is on the graph of this function.

Find the Behavior (Leading Coefficient Test)

Determine the end behavior of the function

f (x) = - 3 x^{4} + 2 x^{2} - 1

using the Leading Coefficient Test

Determining Odd and Even Functions

Determine whether the function

f (x) = x^{4} - 2 x^{2} + 1

is odd, even, or neither.

Finding the Symmetry

Determine whether the function

f (x) = x^{4} - 4 x^{2} + 1

is symmetric about the y-axis, the origin, or neither.

Finding the Asymptotes

Find the vertical and horizontal asymptotes of the function

f (x) = \frac{3 x^{2} - 2 x + 1}{x^{2} - 1}

.

Difference Quotient

Find the difference quotient for the function

f (x) = 3 x^{2} - 2 x + 1

.

Finding the Antiderivative

Find the antiderivative of the function

f (x) = 3 x^{2} - 5 x + 7

.

Graphing

Find the maximum and minimum points on the graph of the function

f (x) = x^{3} - 6 x^{2} + 9 x + 15

.

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