Problem 79
Question
Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$f(x)=\frac{x^{4}}{4}$$
Step-by-Step Solution
Verified Answer
The function \(f(x) = \frac{x^4}{4}\) is a one-to-one function and has an inverse that is also a function. This confirms that the function passes the horizontal line test.
1Step 1: Understanding the Function and Graphing it
The function \(f(x) = \frac{x^4}{4}\) is a polynomial function of even degree. When plotted on a graphical calculator, it gives a U-shaped curve which is above the x-axis.
2Step 2: Check if the Function has an Inverse
As we noted from the graph, the function \(f(x) = \frac{x^4}{4}\) passes the horizontal line test. Therefore, we can say that it is one-to-one, and it has an inverse that is also a function.
3Step 3: Conclusion
From graphing the function and testing it for one-to-one correspondence, it can be concluded that the function \(f(x) = \frac{x^4}{4}\) has an inverse that is a function. The function is one-to-one, meaning for each x-value, there is only one corresponding y-value.
Key Concepts
Graphing Polynomial FunctionsHorizontal Line TestInverse FunctionsPolynomial Functions
Graphing Polynomial Functions
When encountering a polynomial function like the one in our exercise, graphing is a key step to visualize and understand its behavior. Polynomial functions are made up of terms consisting of a variable raised to a non-negative integer power, multiplied by a coefficient. The most basic of these is a linear function, but as the power increases, their shapes become more complex.
Polymer functions with even powers, such as the given function \(f(x) = \frac{x^4}{4}\), tend to display symmetry and have a 'U' shape (for positive leading coefficients) or an 'n' shape (for negative leading coefficients). To graph such a function, you can plot several key points by substituting values for x into the function to find corresponding y-values. Connect these points smoothly, keeping in mind the symmetry and end behavior of the polynomial. For a positive even-degree polynomial like this one, the ends of the graph will rise to infinity.
Polymer functions with even powers, such as the given function \(f(x) = \frac{x^4}{4}\), tend to display symmetry and have a 'U' shape (for positive leading coefficients) or an 'n' shape (for negative leading coefficients). To graph such a function, you can plot several key points by substituting values for x into the function to find corresponding y-values. Connect these points smoothly, keeping in mind the symmetry and end behavior of the polynomial. For a positive even-degree polynomial like this one, the ends of the graph will rise to infinity.
Horizontal Line Test
The horizontal line test is a simple yet effective method for determining if a function is one-to-one, a characteristic required for a function to have an inverse that is also a function. To administer this test, imagine drawing horizontal lines across the graph of the function. If any horizontal line crosses the graph more than once, the function is not one-to-one.
In our case, the function \(f(x) = \frac{x^4}{4}\) is a one-to-one function, as any horizontal line would intersect the graph at most one time. This indicates that for every y-value, there is only one unique x-value. As Step 2 of the provided solution points out, passing this test confirms that the function's inverse will also be a function.
In our case, the function \(f(x) = \frac{x^4}{4}\) is a one-to-one function, as any horizontal line would intersect the graph at most one time. This indicates that for every y-value, there is only one unique x-value. As Step 2 of the provided solution points out, passing this test confirms that the function's inverse will also be a function.
Inverse Functions
When we talk about inverse functions, we refer to a pair of functions that essentially 'undo' each other. For every function \(f\), an inverse function \(f^{-1}\) takes the output of \(f\) and returns it back to its original input value. However, not all functions have inverses that are themselves functions. For a function to have an inverse that is also a function, it must be one-to-one.
A real-life example of inverse functions is converting between Celsius and Fahrenheit temperatures. The process of converting from Celsius to Fahrenheit and then back again (using the corresponding inverse formula) demonstrates how inverse functions operate. In our exercise, the fact that the graph of \(f(x) = \frac{x^4}{4}\) passes the horizontal line test indicates that it does indeed have an inverse function, which is not always the case for polynomial functions.
A real-life example of inverse functions is converting between Celsius and Fahrenheit temperatures. The process of converting from Celsius to Fahrenheit and then back again (using the corresponding inverse formula) demonstrates how inverse functions operate. In our exercise, the fact that the graph of \(f(x) = \frac{x^4}{4}\) passes the horizontal line test indicates that it does indeed have an inverse function, which is not always the case for polynomial functions.
Polynomial Functions
Polynomial functions form a foundational concept in algebra and appear in various forms across mathematical applications. A polynomial function is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
Characteristics of polynomial functions include their degree, which dictates their graph's general shape, and their leading coefficient, which affects the graph's orientation. Taking the function from our exercise \(f(x) = \frac{x^4}{4}\), the highest power is 4, which makes it a quartic function. Polynomial functions of even degrees tend to have similar end behaviors—either both ends up or both ends down, depending on whether the leading coefficient is positive or negative. In our exercise, the leading coefficient is positive, which aligns with the 'U' shape we discussed earlier. Each polynomial function has a unique set of x-intercepts (roots) and y-intercepts, and its graph’s curvature changes at points called turning points, which can be fewer in number than the degree of the polynomial.
Characteristics of polynomial functions include their degree, which dictates their graph's general shape, and their leading coefficient, which affects the graph's orientation. Taking the function from our exercise \(f(x) = \frac{x^4}{4}\), the highest power is 4, which makes it a quartic function. Polynomial functions of even degrees tend to have similar end behaviors—either both ends up or both ends down, depending on whether the leading coefficient is positive or negative. In our exercise, the leading coefficient is positive, which aligns with the 'U' shape we discussed earlier. Each polynomial function has a unique set of x-intercepts (roots) and y-intercepts, and its graph’s curvature changes at points called turning points, which can be fewer in number than the degree of the polynomial.
Other exercises in this chapter
Problem 79
How is the standard form of a circle's equation obtained from its general form?
View solution Problem 79
Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=2 x^{2}+x-1$$
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Express the given function \(h\) as a composition of two functions \(f\) and \(g\) so that \(h(x)=(f \circ g)(x)\). $$h(x)=|2 x-5|$$
View solution Problem 79
Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$g(x)=2 \sqrt{x+2}-2$$
View solution