What is the range of this absolute value function?

An absolute value function, denoted as |x|, represents the distance between a number and zero on a number line without regard to the direction. When dealing with an absolute value function, the range refers to the set of possible values that the function outputs. To determine the range of an absolute value function, it’s important to understand its behavior and characteristics.

The range of an absolute value function depends on the coefficients and constants in the equation. Let’s take a closer look at the different scenarios and how they impact the range:

1. What is the range of the function f(x) = |x|?

The absolute value function f(x) = |x| has a range of [0, ∞). This implies that the output values can range from zero to positive infinity.

2. What is the range of the function f(x) = 3|x|?

The coefficient in front of the absolute value, in this case, being 3, stretches or compresses the graph vertically. The range of f(x) = 3|x| is [0, ∞) since the function will still output values between zero and positive infinity.

3. What is the range of the function f(x) = -2|x|?

The coefficient -2 reflects the graph of the absolute function over the x-axis and vertically compresses the graph. The range of f(x) = -2|x| is (-∞, 0] since this function will give outputs from negative infinity to zero.

4. What is the range of the function f(x) = |x| + 2?

The constant term “+2” translates the graph vertically by two units upwards. This means the range of f(x) = |x| + 2 is [2, ∞), as the output values start from 2 and go up to infinity.

5. What is the range of the function f(x) = |x – 3|?

When there is a constant inside the absolute value function, as in f(x) = |x – 3|, it represents a horizontal shift. The range of this function would be [0, ∞), indicating output values from zero to positive infinity.

6. What is the range of the function f(x) = |3x|?

The coefficient 3 stretches or compresses the graph horizontally. However, it does not affect the range, which remains [0, ∞).

7. What is the range of the function f(x) = |x| – 5?

When there is a constant term inside the absolute value function, a vertical shift occurs. The range of f(x) = |x| – 5 is [-5, ∞), meaning output values start from -5 and go up to positive infinity.

8. What is the range of the function f(x) = 2|x| – 3?

This absolute value function involves both a horizontal stretch due to the coefficient 2 and a vertical shift due to the constant term -3. The range of f(x) = 2|x| – 3 is [-3, ∞), indicating output values from -3 to positive infinity.

9. What is the range of the function f(x) = |x – 2| + 4?

A constant inside the absolute value function represents a horizontal shift. Therefore, the range of f(x) = |x – 2| + 4 is [4, ∞), meaning output values start from 4 and go up to positive infinity.

10. What happens to the range when the coefficient is negative?

If the coefficient of the absolute value function is negative, such as f(x) = -|x|, the graph is reflected over the x-axis, resulting in a range of (-∞, 0].

11. Can the range of an absolute value function be an empty set?

No, the range of an absolute value function is always non-empty because the absolute value of any real number is non-negative, including zero.

12. Does the range change if there is an additional term outside the absolute value?

No, the range is unaffected by terms outside the absolute value. They only lead to vertical shifts. The range is determined solely by the absolute value function itself.

In conclusion, the range of an absolute value function depends on the coefficients and constants involved, but it is always of the form [a, ∞) or (-∞, a], where “a” is a real number that represents the lower or upper bound of the range.

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