Graph Analysis and Function Behavior

Fill in the blank: The function f(x) = (1/3)x^(-2) is _______ at x = 0.

• continuous

• discontinuous

• increasing

• decreasing

Answer explanation

The function f(x) = (1/3)x^(-2) is undefined at x = 0, leading to a discontinuity. Therefore, it is correct to say that the function is discontinuous at x = 0.

Tags

CCSS.HSF-IF.C.7D

At which interval is the function f(x) = (1/3)x^(-2) increasing?

• (-∞, 0)

• (0, ∞)

• (-∞, ∞)

• None

Answer explanation

The function f(x) = (1/3)x^(-2) is defined for x > 0. Its derivative f'(x) = -(2/3)x^(-3) is negative for x < 0 and positive for x > 0, indicating that f(x) is increasing on the interval (0, ∞). Thus, the correct answer is (0, ∞).

What is the behavior of the function f(x) = (1/3)x^(-2) as x approaches infinity?

• Increases

• Decreases

• Approaches zero

• Approaches infinity

Answer explanation

As x approaches infinity, the term x^(-2) approaches zero because the negative exponent indicates a reciprocal. Therefore, f(x) = (1/3)x^(-2) approaches zero.

Tags

CCSS.HSF-IF.C.7E

What is the behavior of the function f(x) = (1/3)x^(-2) as x approaches zero from the right?

• Increases

• Decreases

• Approaches zero

• Approaches infinity

Answer explanation

As x approaches zero from the right, f(x) = (1/3)x^(-2) becomes very large because the negative exponent indicates that the function increases without bound. Therefore, f(x) approaches infinity.

Tags

CCSS.HSF-IF.C.7E

Describe the end behavior of the graph of f(x) = -2x^4 - x^3 + 4x^2 - 2x + 8 using limits.

• lim x→∞ f(x) = ∞ and lim x→-∞ f(x) = -∞

• lim x→∞ f(x) = -∞ and lim x→-∞ f(x) = ∞

• lim x→∞ f(x) = -∞ and lim x→-∞ f(x) = -∞

• lim x→∞ f(x) = ∞ and lim x→-∞ f(x) = ∞

Answer explanation

For f(x) = -2x^4 - x^3 + 4x^2 - 2x + 8, the leading term -2x^4 dominates as x approaches ±∞. Thus, lim x→∞ f(x) = -∞ and lim x→-∞ f(x) = ∞. The correct choice is: lim x→∞ f(x) = -∞ and lim x→-∞ f(x) = ∞.

Fill in the blank: The domain of a function is the set of all possible input values, while the range is the set of all possible ______ values.

• output

• input

• real

• complex

Answer explanation

The domain of a function includes all possible input values, while the range consists of all possible output values. Therefore, the correct choice to fill in the blank is 'output'.

Tags

CCSS.8.F.A.1

CCSS.HSF.IF.A.1

What are the turning points of the function f(x) = x^4 - 2x^2 + 1?

• 2 turning points

• 3 turning points

• 4 turning points

• No turning points

Answer explanation

To find the turning points of f(x) = x^4 - 2x^2 + 1, we calculate the derivative f'(x) = 4x^3 - 4x. Setting f'(x) = 0 gives x = 0, ±1. Evaluating the second derivative shows that there are 2 local minima, confirming 2 turning points.

• (x - 5)

• (x - 2)

• (x + 3)

• (x - 1)

Answer explanation

To determine which binomial is not a factor of the polynomial, we can use the Rational Root Theorem or synthetic division. Testing (x + 3) shows it does not yield a remainder of zero, confirming it is not a factor.

Tags

CCSS.HSA.APR.B.2

Complete the factorization of x^4 - 2x^3 - 33x^2 + 50x + 200 using the factor x^2 - 25.

Answer explanation

To factor x^4 - 2x^3 - 33x^2 + 50x + 200 using x^2 - 25, we first recognize that x^2 - 25 = (x - 5)(x + 5). Dividing the polynomial by x^2 - 25 gives us a quadratic factor, leading to the complete factorization: (x^2 - 25)(x^2 + 5x + 8).

Tags

CCSS.HSA.APR.B.2

• x^3 + x^2 + 2

• x^3 + x^2 - 3

• x^3 + 2x^2 - 3

• x^3 + 2x^2 + 1

Answer explanation

To factor the polynomial, we can use synthetic division or polynomial long division with the root x = -1. Dividing gives us x^3 + x^2 - 3, confirming that (x+1)(x^3 + x^2 - 3) equals the original polynomial.

Tags

CCSS.HSA.APR.B.2