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Chapter 2: Problem 1

Let \(f(x)=x^{2}\) and \(g(x)=2 x-5 .\) Match each function in Group I with the correct expression in Group II. $$I$$ $$(f+g)(x)$$ $$\mathbf{II}$$ $$A.\quad4 x^{2}-20 x+25$$ $$B.\quad x^{2}-2 x+5$$ $$C.\quad 2 x^{2}-5$$ $$D.\quad \frac{x^{2}}{2 x-5}$$ $$E. \quad x^{2}+2 x-5$$ $$F. \quad 2 x^{3}-5 x^{2}$$

Short Answer

Expert verified
(f+g)(x) matches with option E: \(x^2 + 2x - 5\).

Step by step solution

01

Understand the Problem

We need to find an expression for \((f+g)(x)\) where \(f(x) = x^2\) and \(g(x) = 2x - 5\). This means adding the functions \(f(x)\) and \(g(x)\) together.
02

Substitute the Functions

The expression \((f+g)(x)\) means we need to substitute \(f(x) = x^2\) and \(g(x) = 2x - 5\). Therefore, \((f+g)(x)\) becomes \((x^2) + (2x - 5)\).
03

Simplify the Expression

Now, combine like terms: \(x^2 + 2x - 5\). This is the simplified form of \((f+g)(x)\).
04

Match with Group II

Compare the expression \(x^2 + 2x - 5\) with the expressions in Group II. This matches with option E: \(x^2 + 2x - 5\).

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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 fundamental in understanding algebra. They are composed of a sum of terms, each consisting of a variable raised to a power and multiplied by a coefficient. In general, a polynomial function can be expressed as follows:
  • The basic form: \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)
  • Each term, like \( a_ix^i \), represents a part of the polynomial function.
  • The highest power of the variable is called the degree of the polynomial.
In our exercise, the polynomial function \( f(x) = x^2 \) is a simple quadratic polynomial with a degree of 2 since the highest power is 2. This simplicity helps in learning basic operations like function addition, where such polynomials are combined with other types of functions, like linear functions. Understanding this core structure is essential for progressing in algebra.
Expressions Simplification
Simplifying expressions in mathematics means reducing them to their simplest form, which involves combining like terms and performing any possible arithmetic operations. In the context of the given exercise, simplification took place when we combined the terms from \( f(x) \) and \( g(x) \):
  • First, list out all terms: \( x^2, 2x, \text{ and } -5 \).
  • No further like terms can be combined since they all have different degrees.
  • The result, \( x^2 + 2x - 5 \), is the simplest form of the function addition \((f+g)(x)\).
Simplification is key for solving equations and functions because it presents the expression in a more manageable form, making it easier to work with or compare with other mathematical expressions. The goal is to make mathematics clearer and more intuitive, especially when performing function operations like in our example.
Function Operations
Function operations include addition, subtraction, multiplication, and division of functions. These operations combine two or more functions to produce a new function, allowing for a broader analysis of mathematical models. In our exercise, the operation in focus is function addition:
  • The addition of two functions \( (f+g)(x) \) involves adding their outputs for any input \( x \): \( f(x) + g(x) \).
  • We take \( f(x) = x^2 \) and \( g(x) = 2x - 5 \), and compute \( (f+g)(x) = x^2 + 2x - 5 \).
  • This operation illustrates how functions can be combined to form a new function that inherits characteristics from each function, such as shape and intercepts.
Function operations allow us to explore complex relationships between variables and serve as tools for modeling and solving real-world problems. Mastering these operations is crucial as they lay the groundwork for more advanced studies in mathematics.

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Most popular questions from this chapter

Solve each application of openations and composition of functions. Dimensions of a Rectangle Suppose that the length of a rectangle is twice its width. Let \(x\) represent the width of the rectangle. (a) Write a formula for the perimeter \(P\) of the rectangle in terms of \(x\) alone. Then use \(P(x)\) notation to describe it as a function. What type of function is this? (b) Graph the function \(P\) as \(Y_{1}\) found in part (a) in the window \([0,10]\) by \([0,100]\). Locate the point for which \(x=4,\) and explain what \(x\) and \(y\) represent. (c) On the graph of \(P\), locate the point with \(x\) -value 4 . Then sketch a rectangle satisfying the conditions described carlier, and evaluate its perimeter if its width is this \(x\) -value. Use the standard perimeter formula. How does the result compare with the y-value shown on your screen? (d) On the graph of \(P\), find a point with an integer y-value. Interpret the \(x\) - and y-coordinates here.

Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-43.5| \leq 8.5,\) Punta Arenas, Chile

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=\sqrt{x^{2}+1}$$

Based on the ordered pairs seen in each table, make a conjecture about whether the finction \(f\) is even, odd, or neither even nor odd. $$\begin{array}{r|r}x & f(x) \\\\-3 & -1 \\\\-2 & 0 \\\\-1 & 1 \\\0 & 2 \\\1 & 3 \\\2 & 4 \\\3 & 5\end{array}$$

Solve each problem. Dr. Cazayoux has found that, over the years, \(95 \%\) of the babies he delivered weighed \(x\) pounds, where \(|x-8.0| \leq 1.5 .\) What range of weights corresponds to this inequality?
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