Problem 53
Question
Determine the domains of (a) \(f,\) (b) \(g\) and (c) \(f \circ g .\) Use a graphing utility to verify your results. $$f(x)=x+2, \quad g(x)=\frac{1}{x^{2}-4}$$
Step-by-Step Solution
Verified Answer
The domain of \(f(x)\) is all real numbers, the domain of \(g(x)\) is all real numbers except -2 and 2, and the domain of \(f(g(x))\) is all real numbers except -2 and 2.
1Step 1: Determine the domain of \(f(x)\)
The function \(f(x) = x + 2\) is a linear function and it does not have any restrictions on its domain. Hence, its domain is all real numbers, represented as \(-\infty < x < \infty\).
2Step 2: Determine the domain of \(g(x)\)
The denominator of the function \(g(x) = \frac{1}{x^{2}-4}\) should not be zero to stay clear of undefined outcomes. Thus, setting \(x^{2} - 4 = 0\), we get \(x = -2\) and \(x = 2\). Therefore, the domain of \(g(x)\) is all real numbers except -2 and 2, it can be represented as \(-\infty < x < -2\) or \(-2 < x < 2\) or \(2 < x < \infty\).
3Step 3: Determine the domain of \(f(g(x))\)
We have to consider the domain of both \(f(x)\) and \(g(x)\). The domain of \(f(x)\) is all real numbers, whereas the domain of \(g(x)\) is all real numbers except -2 and 2. So, the domain of \(f(g(x))\) becomes all real numbers except -2 and 2.
Key Concepts
Linear FunctionsComposition of FunctionsGraphing Utilities
Linear Functions
Linear functions are among the most straightforward mathematical functions, often taking the form of \(f(x) = mx + b\), where \(m\) and \(b\) are constants. The graph of a linear function is a straight line.
This simplicity makes them particularly useful for modeling situations where there is a constant rate of change. For example, if \(x\) represents time, and \(f(x)\) represents distance, then the constant \(m\) indicates speed. In our specific example, the function \(f(x) = x + 2\) is linear.
This means it has no restrictions on the domain. Every real number is a valid input for \(x\). In other words, the domain of \(f(x) = x + 2\) is \(-\infty < x < \infty\). This characteristic is typical of all linear functions.
This simplicity makes them particularly useful for modeling situations where there is a constant rate of change. For example, if \(x\) represents time, and \(f(x)\) represents distance, then the constant \(m\) indicates speed. In our specific example, the function \(f(x) = x + 2\) is linear.
This means it has no restrictions on the domain. Every real number is a valid input for \(x\). In other words, the domain of \(f(x) = x + 2\) is \(-\infty < x < \infty\). This characteristic is typical of all linear functions.
Composition of Functions
Composition of functions involves applying one function to the results of another. If you have two functions, say \(f\) and \(g\), the composition is typically denoted by \(f \circ g(x)\), which means \(f(g(x))\).
Understanding the domain of composed functions requires considering the domains of both functions involved. For our functions, \(f(x) = x + 2\) and \(g(x) = \frac{1}{x^2 - 4}\), the composition \(f(g(x))\) requires that \(g(x)\) itself is defined. Thus, the domain of \(f(g(x))\) is all real numbers where \(x\) is not -2 or 2,- since these make \(g(x)\) undefined.
Analyzing composed functions in this manner ensures that the resulting function is valid and avoids undefined outputs.
Understanding the domain of composed functions requires considering the domains of both functions involved. For our functions, \(f(x) = x + 2\) and \(g(x) = \frac{1}{x^2 - 4}\), the composition \(f(g(x))\) requires that \(g(x)\) itself is defined. Thus, the domain of \(f(g(x))\) is all real numbers where \(x\) is not -2 or 2,- since these make \(g(x)\) undefined.
Analyzing composed functions in this manner ensures that the resulting function is valid and avoids undefined outputs.
Graphing Utilities
Graphing utilities are tools, often software, that help visualize mathematical functions and their behaviors. They are especially helpful in verifying analytical results, such as domain calculations.
By inputting functions like \(f(x)\) and \(g(x)\) into a graphing utility, you can observe the graphs to see where they fail to produce outputs. For instance, attempting to graph \(g(x) = \frac{1}{x^2 - 4}\) would show vertical asymptotes at \(x = -2\) and \(x = 2\), confirming the absence of those values in its domain.
These visual aids are tremendous for understanding nuances you'll only find in a graph and are especially significant when dealing with composed or more complex functions. They validate results discussed theoretically and often reveal additional insights about the function's behavior.
By inputting functions like \(f(x)\) and \(g(x)\) into a graphing utility, you can observe the graphs to see where they fail to produce outputs. For instance, attempting to graph \(g(x) = \frac{1}{x^2 - 4}\) would show vertical asymptotes at \(x = -2\) and \(x = 2\), confirming the absence of those values in its domain.
These visual aids are tremendous for understanding nuances you'll only find in a graph and are especially significant when dealing with composed or more complex functions. They validate results discussed theoretically and often reveal additional insights about the function's behavior.
Other exercises in this chapter
Problem 52
Write an equation of the line that passes through the points. Use the slope- intercept form (if possible). If not possible, explain why and use the general form
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Use a graphing utility to graph the function. State the domain and range of the function. Describe the pattern of the graph. $$s(x)=2\left(\frac{1}{4} x-\left[\
View solution Problem 53
Use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$
View solution Problem 53
Find all values of \(x\) such that \(f(x)=0.\) $$f(x)=15-3 x$$
View solution