Problem 1
Question
Let \(f(x)=x^{2}\) and \(g(x)=1 / x\). Use your knowledge of the graphs of \(f\) and \(g\) to sketch the graph of \(h(x)=f(x) \cdot g(x)\). Where is \(h(x)\) unde ned? How can you indicate this on your graph? How does your graphing calculator deal with the point at which \(h\) is unde ned?
Step-by-Step Solution
Verified Answer
The graph of \(h(x) = x^{2} \cdot 1/x\) or \(h(x) = x\), except at x = 0 where the function is undefined. This can be indicated by a hole or break at x = 0. A graphing calculator may handle this by simply skipping over x = 0.
1Step 1: Function Identification
Identify functions \(f(x)\) and \(g(x)\) as \(f(x)=x^{2}\) and \(g(x)=1 / x\) respectively.
2Step 2: Understanding the product Function
Understand that \(h(x)\) is the product of \(f(x)\) and \(g(x)\). Therefore, \(h(x) = f(x) \cdot g(x) = x^{2} \cdot \frac{1}{x} = x\).
3Step 3: Identify Where Function is Undefined
Function \(g(x)\) is undefined at x = 0. Therefore, \(h(x)\) will also be undefined at x = 0.
4Step 4: Plot the Graphs of \(f(x)\), \(g(x)\) and \(h(x)\)
Plot the graphs of \(f(x) = x^{2}\), \(g(x) = 1/x\) and \(h(x) = x\) in set of axis. The graphs of \(f(x)\) and \(g(x)\) are parabola and hyperbola, respectively. The product, \(h(x)\), is a straight line, except at x = 0 where it is undefined. That point can be indicated on the graph by a hole, break or asymptote at x = 0.
5Step 5: Use a Graphing Calculator
When dealing with the point at which \(h(x)\) is undefined, a graphing calculator may simply skip over the point, showing a straight line except at x = 0.
Key Concepts
Product of FunctionsUndefined PointsGraphing Calculator UsageParabola and Hyperbola
Product of Functions
A product of functions involves multiplying two or more functions together to create a new function. In this exercise, we work with two functions: \( f(x) = x^2 \) and \( g(x) = \frac{1}{x} \).
When you multiply these two, their product is \( h(x) = f(x) \cdot g(x) = x^2 \cdot \frac{1}{x} \).
Simplifying further, this gives \( h(x) = x \), meaning our product function has become a new, simpler function.
Utilizing the properties of the original functions allows us to understand the behavior of their product. When graphing \( h(x) \), you'll generally see a straight line, reflecting the simplified form of the product function. This is a key skill in graphing and analyzing functions.
When you multiply these two, their product is \( h(x) = f(x) \cdot g(x) = x^2 \cdot \frac{1}{x} \).
Simplifying further, this gives \( h(x) = x \), meaning our product function has become a new, simpler function.
Utilizing the properties of the original functions allows us to understand the behavior of their product. When graphing \( h(x) \), you'll generally see a straight line, reflecting the simplified form of the product function. This is a key skill in graphing and analyzing functions.
Undefined Points
An important aspect of dealing with functions and their graphs is understanding where they are undefined. In our example, the function \( g(x) = \frac{1}{x} \) is undefined where \( x = 0 \).
This is due to division by zero, which is mathematically impossible, creating a so-called 'undefined point'.
Consequently, \( h(x) = x \), the product function, is also undefined at \( x = 0 \).
On a graph, these undefined points are often marked with a hole or discontinuity, assisting in visualizing where the function fails to exist. Understanding and handling these points is crucial when sketching or interpreting graphs of functions.
This is due to division by zero, which is mathematically impossible, creating a so-called 'undefined point'.
Consequently, \( h(x) = x \), the product function, is also undefined at \( x = 0 \).
On a graph, these undefined points are often marked with a hole or discontinuity, assisting in visualizing where the function fails to exist. Understanding and handling these points is crucial when sketching or interpreting graphs of functions.
Graphing Calculator Usage
Graphing calculators are wonderful tools for visualizing functions and their behaviors. When sketching \( h(x) = x \) using a calculator, you'll notice the line technically continues through \( x = 0 \).
However, as noted, it is truly undefined at this point.
Many calculators might handle this by simply omitting data for \( x = 0 \) or displaying a gap.
This illustrates the practical side of understanding undefined points and highlights the importance of double-checking results and visually interpreting graphs correctly when using technology.
However, as noted, it is truly undefined at this point.
Many calculators might handle this by simply omitting data for \( x = 0 \) or displaying a gap.
This illustrates the practical side of understanding undefined points and highlights the importance of double-checking results and visually interpreting graphs correctly when using technology.
Parabola and Hyperbola
In the given exercise, while you simplify the product of functions to \( h(x) = x \), the graphs of the individual functions, \( f(x) \) and \( g(x) \), reveal interesting shapes—namely, a parabola and hyperbola, respectively.
A parabola is the shape of the graph \( f(x) = x^2 \) makes, and it opens upwards, symmetrically about the y-axis.
A hyperbola is the shape revealed by \( g(x) = \frac{1}{x} \), consisting of two disconnected curves that reflect about the origin.
Understanding these shapes aids in recognizing graph patterns and behaviors of rational functions, guiding accurate sketching and analysis of more complex function products. By seeing the original curves, you’ll appreciate how they transform or how their products interact, as seen in our linear result, \( h(x) \).
A parabola is the shape of the graph \( f(x) = x^2 \) makes, and it opens upwards, symmetrically about the y-axis.
A hyperbola is the shape revealed by \( g(x) = \frac{1}{x} \), consisting of two disconnected curves that reflect about the origin.
Understanding these shapes aids in recognizing graph patterns and behaviors of rational functions, guiding accurate sketching and analysis of more complex function products. By seeing the original curves, you’ll appreciate how they transform or how their products interact, as seen in our linear result, \( h(x) \).
Other exercises in this chapter
Problem 1
To nd \(f(g(x))\), apply \(g\) to \(x\) and then use the output of \(g\) as the input of \(f .\) Work from the inside out. Let \(f(x)=x^{2}, g(x)=1 / x\), and \
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The zeros of the function \(f(x)\) are at \(x=-4,-1,2\), and 8 . What are the zeros of (a) \(m(x)=5 f(x)\) ? (b) \(g(x)=f(x+2)\) ? (c) \(h(x)=f(2 x)\) ? (d) \(j
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The zeros of the function \(f(x)\) are at \(x=-5,-2,0\), and 5 . Find the zeros of the following functions. If there is not enough information to determine this
View solution Problem 2
Let \(f(x)=|x|\) and \(g(x)=1 / x\). Use your knowledge of the graphs of \(f\) and \(g\) to sketch the graph of \(h(x)=f(x) \cdot g(x)\). Where is \(h(x)\) unde
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