Finding the Range of Functions

The range of a function is the set of all possible output values (y-values).
To find the range, we need to consider the function's behavior, especially its potential maximum and minimum values.

Constant Function: f(x) = c (where c is a constant). The range is {c} because this function only outputs one value.
Linear Function: f(x) = mx + b (where m and b are constants). The range is all real numbers because the line extends infinitely both up and down.
Quadratic Function: f(x) = ax² + bx + c (where a, b, and c are constants). The range depends on the parabola's opening direction, determined by the coefficient 'a':
- If a > 0, the parabola opens upwards, and its minimum value is the vertex's y-coordinate. The range is [vertex's y-coordinate, ∞).
- If a < 0, the parabola opens downwards, and its maximum value is the vertex's y-coordinate. The range is (-∞, vertex's y-coordinate].

Functions involving Square Roots: The expression under the square root must be non-negative (greater than or equal to zero) to produce real outputs. This constraint helps define the range.
Rational Functions (Fractions): Focus on identifying any potential vertical asymptotes (where the denominator becomes zero), which can indicate infinite jumps in the function's output. Pay attention to the behavior of the function as x approaches these asymptotes.
Functions with Combinations of Operations: Combine the strategies above. For example, a function with both a square root and a quadratic term might require analyzing both the square root's non-negativity constraint and the quadratic's opening direction.

f(x) = 5: The range is {5} because the function always outputs 5, regardless of the input.

f(x) = 2x + 5: The range is all real numbers because the function can output any value depending on the input.

f(x) = x² - 2x - 8: The range can be determined by finding the vertex. The vertex is the minimum or maximum point of the parabola. The range will include all values greater than or equal to (or less than or equal to) the y-coordinate of the vertex.
f(x) = 9 - x²: The range is y ≤ 9 because the function is a parabola opening downwards, with a maximum at y = 9.
f(x) = x² - 2: The range is y ≥ -2 because the function is a parabola opening upwards, with a minimum at y = -2.
f(x) = 9 - (x - 2)²: The range is y ≤ 9 because the function is a parabola opening downwards, with a maximum at y = 9.
f(x) = (x - 2)² - 3: The range is y ≥ -3 because the function is a parabola opening upwards, with a minimum at y = -3.

f(x) = 8 / (x - 2): The range is all real numbers except for y = 0 because the function is undefined when x = 2.
f(x) = (x - 4) / (x - 2): The range is all real numbers except for y = 1 because the function has a horizontal asymptote at y = 1.
f(x) = x / (x² + 2): The range is -1/√2 ≤ y ≤ 1/√2 because the function has horizontal asymptotes at y = 0 and y = 1.
f(x) = x² / (x² + 2): The range is 0 ≤ y ≤ 1 because the function has horizontal asymptotes at y = 0 and y = 1.

f(x) = √x + 2 - 4: The range is y ≥ -4 because the function is a square root function which is always greater than or equal to zero, and then shifted down by 4 units.
f(x) = 9 - √4 - x: The range is y ≤ 7 because the function is a square root function which is always greater than or equal to zero, then subtracted from 9, resulting in a maximum value of 7.
f(x) = √16 - x²: The range is 0 ≤ y ≤ 4 because the function is defined only for values of x where 16 - x² ≥ 0, leading to a maximum value of 4.
f(x) = √ - x² - 2x + 3: The range is 0 ≤ y ≤ 2 because the function represents the top half of a circle, with a maximum value of 2.
f(x) = √x² - 16: The range is y ≥ 0 because the function is only defined for values of x where x² - 16 ≥ 0, and the square root of a non-negative number is always non-negative.