Problem 44 - Calculus: One and Several Variables

Question

Sketch the graph and give the domain and range of the function. $$f(x)=\left\\{\begin{array}{cl}x^{2}, & x<0 \\\\-1, & 0

Step-by-Step Solution

Answer

The graph of the piece-wise function consists of a parabola for \(x < 0\), a horizontal line at y = -1 for \(0 < x < 2\), and a straight line of slope 1 for \(2 < x\). The domain of the function is \(x \neq 0, 2\) and the range of the function is \(y \leq 0\), \(y=-1\), and \(y \geq 2\).

Step 1: Analyzing Each Piece

Understand what each piece of the piece-wise function represents. The first piece '\(x^2 \)' is a parabola which is valid for \(x<0\). The second piece '-1' is a constant which is valid for \(0

Step 2: Sketching the Graph

Draw each piece on the same graph but only within its valid interval. For the first piece '\(x^2\)', draw a parabola for \(x<0\). For the second piece '-1', draw a horizontal line at y = -1 for \(0

Step 3: Determining the Domain and Range

The domain of a function is the set of all possible x-values. For this piece-wise function, the domain is \(x \neq 0, 2\) since the function is defined for all \(x < 0\), \(0 < x < 2\), and \(2 < x\). The range of a function is the set of all possible y-values. For this piece-wise function, the range is \(y \leq 0\), \(y=-1\), and \(y \geq 2\).

Key Concepts

Graphing Functions

Graphing a function is a way to visually represent its behavior on a set of axes. This makes it easier to understand how the output, or y-values, change with the input, or x-values.

To graph a function effectively, you need to note key features such as intercepts, slopes, and where the function is not defined. While graphing piece-wise functions like in our example, it's important to sketch each part in its own interval without extending beyond it. Remember, a clear graph can reveal a lot about a function - its continuity, the intervals of increase or decrease, and much more.

Domain and Range

The domain of a function refers to all the possible x-values for which the function is defined, while the range is the set of all potential y-values the function can output. For piece-wise functions, the domain may have breaks, such as our example where the function isn't defined at x=0 and x=2. For the range, we identify the lowest and highest y-values considering all pieces of the function.

To determine domain and range, it helps to picture the graph and pay attention to horizontal and vertical extents. The domain and range give us a complete picture of the inputs a function can accept and the outputs it can produce.

Parabola

A parabola is a U-shaped curve that is the graph of a quadratic function, like the first piece of our function, given by the equation \(y = x^2\). Parabolas open upwards or downwards, depending on the sign of their leading coefficient. In the context of our piece-wise function, the parabola opens upward but we only consider the left side of it since it's defined for \(x < 0\).

Key points to graph a parabola include its vertex, axis of symmetry, and intercepts. For the given function, as x becomes more negative, the y-values increase, forming the characteristic 'arm' of the parabola.

Constant Function

A constant function is a horizontal line on a graph where the y-value remains the same, regardless of the x-value. In the piece-wise function, the section \(y = -1\) for \(0 < x < 2\) represents a constant function. This indicates that throughout this interval, no matter the x-value, the output will always be -1. Constant functions have no slope, and their domain can be any x-value within the defined interval while the range is a single value.

They're instrumental in creating a 'step' effect in piece-wise functions, as observed in our example.

Linear Function

Linear functions create straight lines with a constant slope. The final piece of our piece-wise function is a linear function represented by \(y = x\). For \(2 < x\), the line continues infinitely with a slope of 1, indicating a 45-degree angle with the x-axis. The slope tells us for every unit increase in x, y increases by the same amount.

Linear functions are simple to graph; they only need two points to draw the line completely. Their domain and range are typically all real numbers, unless stipulated by a piece-wise definition as in our example.

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Problem 44

Other exercises in this chapter

Problem 43

Sketch the graph and give the domain and range of the function. $$f(x)=\left\\{\begin{aligned}1+x, & 0 \leq x \leq 1 \\\x, & 1

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Problem 43

Evaluate to four decimal place accuracy. $$\tan 72.4^{\circ}$$.

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Problem 44

State whether the set is bounded above, bounded below, bounded. If a set is bounded above, give an upper bound; if it is bounded below, give a lower bound; if i

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Problem 44

Evaluate to four decimal place accuracy. $$\cos (7.311)$$.

View solution

Key Concepts

Graphing Functions

Graphing a function is a way to visually represent its behavior on a set of axes. This makes it easier to understand how the output, or y-values, change with the input, or x-values.

To graph a function effectively, you need to note key features such as intercepts, slopes, and where the function is not defined. While graphing piece-wise functions like in our example, it's important to sketch each part in its own interval without extending beyond it. Remember, a clear graph can reveal a lot about a function - its continuity, the intervals of increase or decrease, and much more.

Domain and Range

The domain of a function refers to all the possible x-values for which the function is defined, while the range is the set of all potential y-values the function can output. For piece-wise functions, the domain may have breaks, such as our example where the function isn't defined at x=0 and x=2. For the range, we identify the lowest and highest y-values considering all pieces of the function.

To determine domain and range, it helps to picture the graph and pay attention to horizontal and vertical extents. The domain and range give us a complete picture of the inputs a function can accept and the outputs it can produce.

Parabola

A parabola is a U-shaped curve that is the graph of a quadratic function, like the first piece of our function, given by the equation \(y = x^2\). Parabolas open upwards or downwards, depending on the sign of their leading coefficient. In the context of our piece-wise function, the parabola opens upward but we only consider the left side of it since it's defined for \(x < 0\).

Key points to graph a parabola include its vertex, axis of symmetry, and intercepts. For the given function, as x becomes more negative, the y-values increase, forming the characteristic 'arm' of the parabola.

Constant Function

A constant function is a horizontal line on a graph where the y-value remains the same, regardless of the x-value. In the piece-wise function, the section \(y = -1\) for \(0 < x < 2\) represents a constant function. This indicates that throughout this interval, no matter the x-value, the output will always be -1. Constant functions have no slope, and their domain can be any x-value within the defined interval while the range is a single value.

They're instrumental in creating a 'step' effect in piece-wise functions, as observed in our example.

Linear Function

Linear functions create straight lines with a constant slope. The final piece of our piece-wise function is a linear function represented by \(y = x\). For \(2 < x\), the line continues infinitely with a slope of 1, indicating a 45-degree angle with the x-axis. The slope tells us for every unit increase in x, y increases by the same amount.

Linear functions are simple to graph; they only need two points to draw the line completely. Their domain and range are typically all real numbers, unless stipulated by a piece-wise definition as in our example.