Domain, Range, And Graph Of H(x) = ∛(x+4)

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Domain, Range, and Graph of h(x) = ∛(x+4)

Hey guys! Let's dive into analyzing the function h(x) = ∛(x+4). We're going to figure out its domain, sketch its graph using point plotting, and then determine its range. This is a classic example of how we can understand a function's behavior by looking at its equation and visualizing it.

(a) Determining the Domain of h(x) = ∛(x+4)

When we talk about the domain of a function, we're essentially asking: what are all the possible input values (x-values) that we can plug into the function and get a real number output? For the function h(x) = ∛(x+4), we need to consider what restrictions, if any, exist.

Specifically, the crucial part of this function is the cube root, ∛(x+4). Unlike square roots, cube roots can accept any real number as input—positive, negative, or zero. This is because a negative number times itself three times results in a negative number, so we can take the cube root of negatives. In more detail:

  • Cube Roots vs. Square Roots: Remember, square roots (√x) only accept non-negative numbers (x ≥ 0) because we can't take the square root of a negative number and get a real number result. However, cube roots are different. For instance, ∛(-8) = -2, because (-2) * (-2) * (-2) = -8.
  • The Expression Inside the Cube Root: In our function, we have ∛(x+4). The expression inside the cube root is (x+4). Since we can take the cube root of any real number, (x+4) can be any real number. This means there are no restrictions on x.

Therefore, the domain of h(x) = ∛(x+4) is all real numbers. We can express this in several ways:

  • Interval Notation: (-∞, ∞)
  • Set Notation: {x | x ∈ ℝ} (which reads as "the set of all x such that x is an element of the real numbers")

So, no matter what value we choose for x, we can always compute h(x). That's a pretty wide-open playing field for our function!

(b) Graphing h(x) = ∛(x+4) Using Point Plotting

Alright, now let's get visual! Graphing a function helps us see its behavior in a way that an equation alone sometimes can't. Point plotting is a straightforward method: we choose some x-values, calculate the corresponding y-values (h(x)), and then plot these points on a coordinate plane. By connecting the points, we can sketch the graph.

To effectively graph h(x) = ∛(x+4), we should choose x-values that make the expression inside the cube root (x+4) easy to evaluate. This often means picking values that result in perfect cubes (like -8, -1, 0, 1, 8, etc.). Here’s a step-by-step approach to create the graph:

  1. Choose x-values: Let's pick some x-values that make (x+4) a perfect cube:

    • If x = -12, then x + 4 = -8, and ∛(-8) = -2.
    • If x = -5, then x + 4 = -1, and ∛(-1) = -1.
    • If x = -4, then x + 4 = 0, and ∛(0) = 0.
    • If x = -3, then x + 4 = 1, and ∛(1) = 1.
    • If x = 4, then x + 4 = 8, and ∛(8) = 2.

  2. Calculate h(x) values:

    • h(-12) = ∛(-12 + 4) = ∛(-8) = -2
    • h(-5) = ∛(-5 + 4) = ∛(-1) = -1
    • h(-4) = ∛(-4 + 4) = ∛(0) = 0
    • h(-3) = ∛(-3 + 4) = ∛(1) = 1
    • h(4) = ∛(4 + 4) = ∛(8) = 2

  3. Create a table of values:

    x h(x) (x, h(x))
    -12 -2 (-12, -2)
    -5 -1 (-5, -1)
    -4 0 (-4, 0)
    -3 1 (-3, 1)
    4 2 (4, 2)

  4. Plot the points: Plot these points (-12, -2), (-5, -1), (-4, 0), (-3, 1), and (4, 2) on a coordinate plane.

  5. Connect the points: Draw a smooth curve through the points. The graph of h(x) = ∛(x+4) will look like a stretched-out S-shape, passing through the point (-4, 0).

By plotting these points and connecting them smoothly, we get a visual representation of how the function behaves. The graph shows us how the output (h(x)) changes as the input (x) varies.

(c) Determining the Range of h(x) = ∛(x+4) Based on the Graph

Okay, we've got our graph! Now, let's figure out the range of the function. The range is all the possible output values (y-values or h(x)-values) that the function can produce. Looking at the graph we just created, we can make some observations.

Think about it this way: the range is like the vertical "reach" of the graph. Does the graph extend infinitely upwards? Does it extend infinitely downwards? Are there any horizontal boundaries that the graph doesn't cross?

For h(x) = ∛(x+4), we can see the following:

  • Extending Upwards: As x gets very large (goes towards positive infinity), h(x) also gets very large (goes towards positive infinity), though at a slower rate due to the cube root.
  • Extending Downwards: As x gets very small (goes towards negative infinity), h(x) also gets very small (goes towards negative infinity).
  • No Horizontal Boundaries: There are no horizontal lines that the graph approaches but never touches (no horizontal asymptotes). The graph smoothly covers all y-values.

Because the cube root function can produce any real number output, and our graph extends infinitely upwards and downwards, the range of h(x) = ∛(x+4) is all real numbers. Just like with the domain, we can express this in different notations:

  • Interval Notation: (-∞, ∞)
  • Set Notation: {y | y ∈ ℝ} (or {h(x) | h(x) ∈ ℝ})

So, the function h(x) = ∛(x+4) can take any real number as input (domain) and produce any real number as output (range). That’s a pretty versatile function, guys!

In summary, by systematically analyzing the function's equation and visualizing its graph, we’ve successfully determined its domain and range. This approach can be applied to many other functions, giving us a powerful tool for understanding mathematical relationships.