Problem 10
Question
In \(7-10,\) the domain of each function is the set of real numbers. a. Sketch the graph of each function. b. What is the range of each function? $$ g(x)=\left|3-\frac{x}{2}\right|-3 $$
Step-by-Step Solution
Verified Answer
The range of the function is \(\{ y \in \mathbb{R} \mid y \geq -3 \}\).
1Step 1: Identify Key Features of the Function
The given function is a modified absolute value function: \ \[ g(x) = \left| 3 - \frac{x}{2} \right| - 3 \] \ The expression inside the absolute value, \(3 - \frac{x}{2}\), suggests a horizontal transformation. The \(-3\) outside the absolute value indicates a vertical shift downwards by 3 units.
2Step 2: Rewrite the Inner Expression
Consider the implication of \(3 - \frac{x}{2} = 0\): \ \[3 = \frac{x}{2} \Rightarrow x = 6\] \ This means the vertex of the absolute value function is horizontally shifted to \(x = 6\). Substitute \(x = 6\) into \(g(x)\) to find the vertex value: \ g(6) = \left| 3 - \frac{6}{2} \right| - 3 = \left| 3 - 3 \right| - 3 = 0 - 3 = -3.
3Step 3: Plot the Key Points and Sketch the Graph
The vertex of the function is at \((6, -3)\). Since it is an absolute value function, it will have a 'V' shape opening upwards. Evaluate additional points for more accuracy: \ \(x = 4\), \(g(4) = \left| 3 - \frac{4}{2} \right| - 3 = \left| 1 \right| - 3 = -2\) \ \(x = 8\), \(g(8) = \left|3 - \frac{8}{2} \right| - 3 = \left| -1 \right| - 3 = -2\) \ Plot the points \((4, -2), (6, -3), (8, -2)\) on a coordinate plane and draw the V-shaped graph.
4Step 4: Determine the Range of the Function
The lowest point of the graph (the vertex) is at \(y = -3\). Since the 'V' shape of the absolute value function opens upwards, the range of the function is \(\{ y \in \mathbb{R} \mid y \geq -3 \}\).
Key Concepts
Graphing FunctionsDomain and RangeFunction Transformations
Graphing Functions
Graphing functions is a fundamental skill in mathematics that allows you to visually represent mathematical relationships. Graphs can reveal things about a function that are hard to see from an equation alone.
When graphing an absolute value function like \( g(x) = \left|3 - \frac{x}{2}\right| - 3 \), it's crucial to first identify the vertex. For this specific function, the vertex is a point where the V-shape of the graph changes direction. The vertex can be determined by setting the inside of the absolute value expression to zero. In this case, by solving \( 3 - \frac{x}{2} = 0 \), we find the vertex at \( x = 6 \), giving us the point \( (6, -3) \).
Next, plot the vertex on a coordinate grid. To complete the V-shape of the absolute value graph, identify additional points by choosing x-values around the vertex, plug these into the function, and solve for y. For example, at \( x = 4 \) and \( x = 8 \), this gives you the points \( (4, -2) \) and \( (8, -2) \). Finally, draw lines connecting these points, ensuring the lines reflect the characteristic symmetry of the V-shape about the vertex.
When graphing an absolute value function like \( g(x) = \left|3 - \frac{x}{2}\right| - 3 \), it's crucial to first identify the vertex. For this specific function, the vertex is a point where the V-shape of the graph changes direction. The vertex can be determined by setting the inside of the absolute value expression to zero. In this case, by solving \( 3 - \frac{x}{2} = 0 \), we find the vertex at \( x = 6 \), giving us the point \( (6, -3) \).
Next, plot the vertex on a coordinate grid. To complete the V-shape of the absolute value graph, identify additional points by choosing x-values around the vertex, plug these into the function, and solve for y. For example, at \( x = 4 \) and \( x = 8 \), this gives you the points \( (4, -2) \) and \( (8, -2) \). Finally, draw lines connecting these points, ensuring the lines reflect the characteristic symmetry of the V-shape about the vertex.
Domain and Range
Understanding the domain and range of a function is crucial because it tells us the input values the function can accept and the output values it can produce.
The domain of an absolute value function like \( g(x) = \left|3 - \frac{x}{2}\right| - 3 \) is typically all real numbers because you can plug any real number into the function. This is especially true here because there are no denominators, even roots, or logarithms in the original function that might restrict this. Thus, the domain of \( g(x) \) is \( \{x \in \mathbb{R}\} \).
The range, however, is determined by the lowest point on the graph. Since the vertex is the lowest point at \( y = -3 \) and the arms of the V-shape open upwards, all y-values are greater than or equal to -3. This constitutes the range: \( \{y \in \mathbb{R} | y \geq -3\} \). Recognizing these aspects of the graph will help you determine the function's behavior.
The domain of an absolute value function like \( g(x) = \left|3 - \frac{x}{2}\right| - 3 \) is typically all real numbers because you can plug any real number into the function. This is especially true here because there are no denominators, even roots, or logarithms in the original function that might restrict this. Thus, the domain of \( g(x) \) is \( \{x \in \mathbb{R}\} \).
The range, however, is determined by the lowest point on the graph. Since the vertex is the lowest point at \( y = -3 \) and the arms of the V-shape open upwards, all y-values are greater than or equal to -3. This constitutes the range: \( \{y \in \mathbb{R} | y \geq -3\} \). Recognizing these aspects of the graph will help you determine the function's behavior.
Function Transformations
Function transformations are changes that shift, stretch, compress, or reflect the graph of a function. Identifying these transformations can help you easily graph complex functions.
In \( g(x) = \left|3 - \frac{x}{2}\right| - 3 \), the expression \( 3 - \frac{x}{2} \) inside the absolute value causes a horizontal shift. Solving \( 3 = \frac{x}{2} \) gives \( x = 6 \), indicating a horizontal shift to the right. The entire graph of the absolute value function is moved to align its vertex with \( x = 6 \).
The \(-3\) outside of the absolute value denotes a vertical shift. The entire graph is lowered by 3 units on the y-axis. The standard absolute value "V" shape is thus translated downward from the origin to the point \( (6, -3) \).
Understanding transformations means recognizing how the graph changes when different operations are applied, making it easier to plot and interpret any function.
In \( g(x) = \left|3 - \frac{x}{2}\right| - 3 \), the expression \( 3 - \frac{x}{2} \) inside the absolute value causes a horizontal shift. Solving \( 3 = \frac{x}{2} \) gives \( x = 6 \), indicating a horizontal shift to the right. The entire graph of the absolute value function is moved to align its vertex with \( x = 6 \).
The \(-3\) outside of the absolute value denotes a vertical shift. The entire graph is lowered by 3 units on the y-axis. The standard absolute value "V" shape is thus translated downward from the origin to the point \( (6, -3) \).
Understanding transformations means recognizing how the graph changes when different operations are applied, making it easier to plot and interpret any function.
Other exercises in this chapter
Problem 10
In \(3-10,\) find each of the function values when \(\mathrm{f}(x)=4 x\) $$ \mathrm{f}\left(\mathrm{f}^{-1}(\sqrt{2})\right) $$
View solution Problem 10
In \(3-10\) , evaluate each composition for the given values if \(f(x)=3 x\) and \(g(x)=x-2\) $$ g\left(f\left(\frac{2}{3}\right)\right) $$
View solution Problem 11
In \(11-16,\) determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function. $$ \\{(
View solution Problem 11
In \(8-13\) , the domain of \(f(x)=4-2 x\) and of \(g(x)=x^{2}\) is the set of real numbers and the domain of \(h(x)=\frac{1}{x}\) is the set of non-zero real n
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