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Chapter 2: Problem 18

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$(f-g)(-2)$$

Short Answer

Expert verified
The result of \((f-g)(-2)\) is 3.

Step by step solution

01

Find Individual Functions at x = -2

First, we need to evaluate each function at \(x = -2\). Start with \(f(x)\). The function \(f(x) = x^2 + 3x\). Plug in \(-2\) for \(x\):\[f(-2) = (-2)^2 + 3(-2) = 4 - 6 = -2\]Now evaluate \(g(x) = 2x - 1\):\[g(-2) = 2(-2) - 1 = -4 - 1 = -5\].
02

Compute the Difference of Functions at x = -2

Now that we have \(f(-2)\) and \(g(-2)\), compute the difference \((f-g)(-2)\):\[(f-g)(-2) = f(-2) - g(-2)\]Substitute the values we found:\[ (f-g)(-2) = (-2) - (-5) = -2 + 5 = 3\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions
Polynomial functions are mathematical expressions involving terms that are powers of a variable, usually denoted as \( x \). The term "polynomial" refers to the sum of more than one monomial. Each monomial is a product of a constant (known as the coefficient) and a non-negative integer power of \( x \). The general form of a polynomial can be expressed as:
\[a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\]
  • Here, \( a_n, a_{n-1}, \ldots, a_0 \) are constants, and \( n \) is a non-negative integer known as the degree of the polynomial.
  • The degree of the polynomial is determined by the highest power of \( x \) in the polynomial.
  • A polynomial with a degree of 0 is a constant function, a degree of 1 is linear, a degree of 2 is quadratic, and so on.
In our problem, \( f(x) = x^2 + 3x \) is a quadratic polynomial because the highest power of \( x \) is 2. Understanding different types of polynomial functions allows you to recognize patterns and solve equations more efficiently.
Difference of Functions
The difference of functions involves subtracting one function from another. When given two functions, \( f(x) \) and \( g(x) \), the difference \((f-g)(x)\) represents a new function.
This operation combines the individual values of \( f(x) \) and \( g(x) \) to produce new results:
\[(f-g)(x) = f(x) - g(x)\]When calculating in practice, it's important to handle signs carefully:
  • Subtracting a negative, like in \( -(-5) \), actually turns into addition: \(-(-5) = +5\).
  • The result of this operation is dependent on the values of \( f(x) \) and \( g(x) \) at specific points.
In the exercise, after evaluating the functions at \( x = -2 \), we found:
  • \( f(-2) = -2 \)
  • \( g(-2) = -5 \)
  • Thus, the difference is \((f-g)(-2) = -2 - (-5) = 3\).
Evaluate Functions
Evaluating functions is determining the output value of a function for a specific input value of \( x \). This involves substituting the input value directly into the function and performing any necessary arithmetic operations.
  • For each function, you replace \( x \) with the given number and simplify.
  • This process reveals how the function behaves at particular points.
  • Careful attention to arithmetic will ensure accurate results.
In our exercise, we evaluated two functions:
\( f(x) = x^2 + 3x \)
and
\( g(x) = 2x - 1 \).
For \( x = -2 \):
  • \( f(-2) = (-2)^2 + 3(-2) = 4 - 6 = -2 \)
  • \( g(-2) = 2(-2) - 1 = -4 - 1 = -5 \)
By evaluating each function, we can then perform operations such as finding the difference, as demonstrated in the exercise.

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Most popular questions from this chapter

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=4 x-\frac{1}{x}$$

Solve each application of openations and composition of functions. Volume of a Sphere The formula for the volume of a sphere is \(V=\frac{4}{3} \pi r^{3},\) where \(r\) represents the radius of the sphere. (a) Write a function \(D(r)\) that gives the volume gained when the radius of a sphere of \(r\) inches is increased by 3 inches. (b) Graph \(y=D(r)\) found in part (a), using \(x\) for \(r,\) in the window \([0,10]\) by \([0,1500]\) (c) Use your calculator to graphically find the amount of volume gained when a sphere with a 4 -inch radius is increased to a 7 -inch radius. (d) Verify your result in part (c) analytically.

The greenhouse gas methane lets sunlight into the atmosphere, but blocks heat from escaping the earth's atmosphere. Methane is a by-product of buming fossil fuels. In the table, \(f\) models the predicted methane emissions in millions of tons produced by developed countries during year \(x\). The function \(g\) models the same emissions for developing countries. $$\begin{array}{|c|c|c|c|c|c|}\hline x & 1990 & 2000 & 2010 & 2020 & 2030 \\\\\hline f(x) & 27 & 28 & 29 & 30 & 31 \\\\\hline g(x) & 5 & 7.5 & 10 & 12.5 & 15\end{array}$$ (a) Make a table for a function \(h\) that models the total predicted methane emissions for developed and developing countries. (b) Write an equation that relates \(f(x), g(x),\) and \(h(x)\)

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=\frac{1}{x+1}, g(x)=5 x$$

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=x^{4}-7 x^{2}+6$$
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