Exploring Power Functions and Graphs

What is the graph of the function f(x) = x^2?

• The graph is a straight line with a slope of 1.

• The graph is a parabola opening downwards with vertex at (0,0).

• The graph is a circle centered at (0,0).

• The graph is a parabola opening upwards with vertex at (0,0).

How does the graph of f(x) = x^(-1) behave as x approaches 0?

• The graph approaches +∞ as x approaches 0 from the right and -∞ as x approaches 0 from the left.

• The graph approaches +∞ from the left and -∞ from the right.

• The graph approaches 0 from both sides as x approaches 0.

• The graph remains constant as x approaches 0.

Identify the end behavior of the function f(x) = x^(3/2).

• f(x) approaches positive infinity as x approaches negative infinity; approaches zero for positive x.

• f(x) approaches zero as x approaches positive infinity; undefined for negative x.

• f(x) approaches positive infinity as x approaches positive infinity; undefined for negative x.

• f(x) approaches negative infinity as x approaches positive infinity; undefined for positive x.

Sketch the graph of f(x) = x^(1/3) and describe its key features.

• The graph has a sharp corner at the origin and does not extend infinitely.

• The graph is a parabola that opens upwards and does not pass through the origin.

• The graph of f(x) = x^(1/3) is a straight line that only extends to the right.

• The graph of f(x) = x^(1/3) is a smooth curve that passes through the origin, is symmetric about the origin, and extends infinitely in both directions.

What happens to the value of f(x) = x^(1/4) as x approaches infinity?

• f(x) approaches zero as x approaches infinity.

• f(x) approaches infinity as x approaches infinity.

• f(x) remains constant as x approaches infinity.

• f(x) oscillates between values as x approaches infinity.

Determine the asymptotic behavior of f(x) = x^(-2).

• f(x) remains constant at 1 as x approaches infinity.

• f(x) oscillates between -1 and 1 as x approaches infinity.

• f(x) approaches infinity as x approaches infinity.

• f(x) approaches 0 as x approaches infinity.

How does the graph of f(x) = x^(5/2) compare to f(x) = x^(2)?

• f(x) = x^(2) grows faster than f(x) = x^(5/2) for all x.

• Both functions are defined for all real x and grow at the same rate.

• f(x) = x^(5/2) grows faster than f(x) = x^(2) for x > 0 and is undefined for negative x.

• f(x) = x^(5/2) is undefined for all x.

What is the y-intercept of the function f(x) = x^(3/4)?

• 2

• 0

• -1

• 1

Describe the behavior of f(x) = x^(-3/2) as x approaches negative infinity.

• f(x) approaches negative infinity as x approaches negative infinity.

• f(x) approaches 1 as x approaches negative infinity.

• f(x) approaches 0 from the negative side as x approaches negative infinity.

• f(x) remains constant at 0 as x approaches negative infinity.

Sketch the graph of f(x) = x^(1/2) and identify its domain.

• Domain: (-∞, 0)

• Domain: (0, 1)

• Domain: [1, ∞)

• Domain: [0, ∞)