Problem 17 - Algebra 2

Question

Graph each function. Identify the domain and range. \(h(x)=-3[x]\)

Step-by-Step Solution

Answer

Domain: all real numbers; Range: all integer multiples of -3.

Step 1: Understand the Function

The function given is a floor function defined as \(h(x) = -3[x]\). In this context, \([x]\) represents the greatest integer less than or equal to \(x\). To understand this, we need to realize that the function scales this integer value by -3.

Step 2: Determine Points Using the Function

Evaluate \(h(x)\) for a few integer and non-integer inputs. For example: - If \(x = 1.5\), then \([1.5] = 1\) and \(h(1.5) = -3(1) = -3\).- If \(x = -2.7\), then \([-2.7] = -3\) and \(h(-2.7) = -3(-3) = 9\). Continue this for values around zero and integers to see patterns.

Step 3: Sketch the Graph

Plot the points found in Step 2 on a coordinate plane. The function \(h(x)\) produces flat segments or steps that change at every integer \(x\). The graph of \(h(x)\) should be a series of horizontal lines, each spanning between integers and changing height every time \(x\) crosses an integer.

Step 4: Identify the Domain

The domain of \(h(x) = -3[x]\) is all real numbers. This is because there are no restrictions on the input \(x\); \(x\) can be any real number.

Step 5: Identify the Range

The range of \(h(x)\) is the set of all outputs that the function can produce. Since \([x]\) generates all integers and each integer is then multiplied by -3, the range is all integer multiples of -3: \{ \ldots, -9, -6, -3, 0, 3, 6, 9, \ldots \}.

Key Concepts

Domain and Range

In the context of functions, the domain is about what values can be fed into a function. It represents all the possible inputs. The range, on the other hand, signifies all the possible outputs or results that a function can yield. For the function given, \( h(x) = -3[x] \), the domain is all real numbers. This means any real number can be substituted into the function.

• The domain: All real numbers (\( \mathbb{R} \))
• The range: All integer multiples of -3 such as \{ \ldots, -9, -6, -3, 0, 3, 6, 9, \ldots \}

Every real number input for \(x\) will have a corresponding output that's an integer multiple of -3. This is due to how the floor function and multiplication by -3 operate together.

Greatest Integer Function

The greatest integer function is commonly referred to as the floor function. It is denoted by \([x]\), symbolizing the greatest integer less than or equal to \(x\). This means that no matter what real number you input, the result will be the closest integer below that number.
For example:

• For 1.5, \([1.5] = 1\)
• For -2.7, \([-2.7] = -3\)

This concept directly affects the function \( h(x) = -3[x] \). It rounds any decimal input down to the nearest whole number before multiplying by -3. Thus:

• \(h(1.5)\) becomes \(-3 \times 1 = -3\)
• \(h(-2.7)\) results in \(-3 \times (-3) = 9\)

The greatest integer function is essential in understanding how outputs are computed in piecewise functions like \(-3[x]\).

Piecewise Linear Function

A piecewise linear function consists of multiple linear segments, each defined over a specific interval of the domain. For \( h(x) = -3[x] \), the graph is not a straight line, but a series of constant horizontal lines. Each segment corresponds to an interval between consecutive integers.
Here's why it's piecewise linear:

• The function value remains constant for any \(x\) within each interval \([n, n+1)\), where \(n\) is an integer.
• The height changes at every integer \(x\), marking the transition to the next segment.

This manifests as a step-like pattern on the graph, showing clearly defined jumps or steps at integer values of \(x\). These horizontal segments make the output flat between each integer step.
The function is instructive for learning how mathematical transformations like the floor function lead to the characteristic step pattern, multiplying every floor output by -3. It results in recognizable visual trends useful in graph analysis.