Problem 30
Question
Let \(f(x)=x^{2}+3 x,\) and let \(g(x)=f(x)+2\) (a) Write the rule of \(g(x)\) (b) Find the difference quotients of \(f(x)\) and \(g(x) .\) How are they related?
Step-by-Step Solution
Verified Answer
Answer: The relationship between the difference quotients of the functions \(f(x)\) and \(g(x)\) is that they are the same, indicating that the average rate of change for both functions is identical.
1Step 1: Part (a): Find the rule of g(x)
We know g(x) is given by the expression \(g(x) = f(x) + 2\). We are given the expression for f(x), so to find g(x), we need to substitute f(x) into the expression.
\(g(x) = f(x) + 2\)
From the exercise, we know that \(f(x) = x^2 + 3x\). So,
\(g(x) = (x^2 + 3x) + 2\)
Simplifying this, we have:
\(g(x) = x^2 + 3x + 2\)
This is the rule of function g(x).
2Step 2: Part (b): Find the difference quotients of f(x) and g(x)
The difference quotient is a measure of the average rate of change of a function. It is defined as:
\(\frac{f(x + h) - f(x)}{h}\)
Let's start with finding the difference quotient of f(x).
\(\frac{f(x + h) - f(x)}{h} = \frac{(x+h)^2 + 3(x+h) - (x^2 + 3x)}{h}\)
Expanding this expression and simplifying it, we get:
\(\frac{(x^2 + 2xh + h^2 + 3x + 3h) - (x^2 + 3x)}{h} = \frac{2xh + h^2 + 3h}{h}\)
Now, let's factor h from the numerator:
\(\frac{h(2x + h + 3)}{h}\)
Since h ≠ 0, we can cancel out h from both numerator and denominator:
\(2x + h + 3\)
Now, let's find the difference quotient for g(x).
\(\frac{g(x + h) - g(x)}{h} = \frac{((x+h)^2 + 3(x + h) + 2) - (x^2 + 3x + 2)}{h}\)
Expanding and simplifying the expression, we find:
\(\frac{(x^2 + 2xh + h^2 + 3x + 3h + 2) - (x^2 + 3x + 2)}{h} = \frac{2xh + h^2 + 3h}{h}\)
This is the same expression we got while finding the difference quotient for f(x). So, we can simplify it the same way:
\(\frac{h(2x + h + 3)}{h} = 2x + h + 3\)
3Step 3: Part (b): Relationship between difference quotients
We find that the difference quotients of \(f(x)\) and \(g(x)\) are the same:
\(f(x): 2x + h + 3\)
\(g(x): 2x + h + 3\)
So, the relationship between the difference quotients of both functions is that they are the same. This implies that the average rate of change for both functions is identical.
Key Concepts
FunctionsPolynomial FunctionsRate of Change
Functions
Understanding functions is crucial in math. They're like machines that take an input, do something with it, and then produce an output.
Functions can be written in many forms, like tables, graphs, or equations. In most math problems, they are expressed using letters such as \(f(x)\), where \(x\) is the input and \(f(x)\) is the output.
Functions must be consistent in their operation. If you put in the same input, you’ll always get the same output. This makes them predictable and reliable for mathematical calculations. For example, with the function \(f(x) = x^2 + 3x\), if you put \(x = 2\) as the input, you will always get the same result, which is 10. In the given exercise, understanding how \(f(x)\) works helps us to define \(g(x)\), which is just a slight modification of it. Here's a handy tip:
Functions can be written in many forms, like tables, graphs, or equations. In most math problems, they are expressed using letters such as \(f(x)\), where \(x\) is the input and \(f(x)\) is the output.
Functions must be consistent in their operation. If you put in the same input, you’ll always get the same output. This makes them predictable and reliable for mathematical calculations. For example, with the function \(f(x) = x^2 + 3x\), if you put \(x = 2\) as the input, you will always get the same result, which is 10. In the given exercise, understanding how \(f(x)\) works helps us to define \(g(x)\), which is just a slight modification of it. Here's a handy tip:
- To evaluate a function at any given \(x\), simply substitute \(x\) in the function’s equation. This will help you find what the output, \(f(x)\), will be.
Polynomial Functions
Polynomial functions are among the most common types of functions you’ll encounter in algebra.
They involve sums of terms with variables raised to power, like \(x^2\), \(x^3\), etc., and coefficients.
The simplest polynomial is a linear equation, like \(ax+b\), but they can grow more complex, such as \(ax^n + bx^{n-1} + ... + k\). Our function \(f(x) = x^2 + 3x\) is a second-degree polynomial, which means it has the highest degree of two. These are also known as quadratic functions. They form parabolic shapes when graphed and are very useful in modeling a variety of real-world phenomena. Another key aspect to note about polynomial functions is:
They involve sums of terms with variables raised to power, like \(x^2\), \(x^3\), etc., and coefficients.
The simplest polynomial is a linear equation, like \(ax+b\), but they can grow more complex, such as \(ax^n + bx^{n-1} + ... + k\). Our function \(f(x) = x^2 + 3x\) is a second-degree polynomial, which means it has the highest degree of two. These are also known as quadratic functions. They form parabolic shapes when graphed and are very useful in modeling a variety of real-world phenomena. Another key aspect to note about polynomial functions is:
- They have smooth and continuous graphs. This means no breaks or holes in the graph, making them very predictable and easy to work with.
Rate of Change
The rate of change helps us understand how a function behaves as its input changes. It's like asking, "How fast is this function changing?"
The difference quotient is a tool for finding the average rate of change of a function between two points.
It's calculated as \(\frac{f(x + h) - f(x)}{h}\), where \(h\) is a small change in \(x\).In our exercise, when we found the difference quotients for \(f(x)\) and \(g(x)\), they were identical: \(2x + h + 3\). This means that despite the small change in \(g(x)\), both functions change at the same rate over any small interval. Some crucial insights about the rate of change are:
The difference quotient is a tool for finding the average rate of change of a function between two points.
It's calculated as \(\frac{f(x + h) - f(x)}{h}\), where \(h\) is a small change in \(x\).In our exercise, when we found the difference quotients for \(f(x)\) and \(g(x)\), they were identical: \(2x + h + 3\). This means that despite the small change in \(g(x)\), both functions change at the same rate over any small interval. Some crucial insights about the rate of change are:
- If the rate of change is constant, as in linear functions, the function’s graph will be a straight line.
- Non-linear functions, such as polynomials, have variable rates of change that make their graph shapes more complex.
Other exercises in this chapter
Problem 29
Could the following statement ever be the rule of a function? For input \(x,\) the output is the number whose square is \(x\) Why or why not? If there is a func
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List three different functions (other than the ones in Example 6 and Exercise 29 ), each of which is its own inverse. [Many correct answers are possible.]
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Verify that \((f \circ g)(x)=x\) and \((g \circ f)(x)=x\) for every \(x\) $$f(x)=\sqrt[3]{x-1}, \quad g(x)=x^{3}+1$$
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Find the approximate location of all local maxima and minima of the function. $$l(x)=\frac{1}{1+x^{2}}$$
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