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Chapter 2: Problem 2

If \(f(x)=3 x^{2}-2 x+1\) and \(g(x)=4 x-1,\) find: a. \((f+g)(x)\) b. \((f+g)(5)\)

Short Answer

Expert verified
a) The sum of the functions, (f+g)(x), is 3x^2 + 2x. b) The value of this combined function when evaluated at x=5 is 85.

Step by step solution

01

Find the sum of the functions

The sum of two functions, (f+g)(x), is the sum of the respective functions. Therefore, add the two functions together to get \( (f+g)(x) = f(x) + g(x) = (3 x^2 - 2 x + 1) + (4 x - 1)\). Simplifying, this results in \( (f+g)(x) = 3x^2 + 2x \).
02

Evaluate the sum of the functions at x=5

Substitute x=5 into the combined function found in step 1, \( (f+g)(5) = 3(5)^{2} + 2(5) \). Simplifying, this results in \( (f+g)(5) = 75 + 10 = 85 \).

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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 expressions composed of variables and coefficients, connected by addition, subtraction, and multiplication, but never division by a variable. These functions are written in the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where each term has a non-negative integer exponent.
  • The highest exponent of the variable is called the degree of the polynomial.
  • In our exercise, \( f(x)=3x^2-2x+1 \) is a quadratic polynomial because the highest exponent of \(x\) is 2.
Polynomial functions can change their appearance based on operations such as addition or subtraction. These characteristics help in modeling various real-world scenarios where relationships can be expressed through equations.
Function Evaluation
Function evaluation refers to the process of finding the value of a function for a specific input. Function evaluation is crucial in mathematics and its applications because it helps in understanding the behavior of functions.
When evaluating a function, follow these steps:
  • Identify the function expression, for instance, \((f+g)(x) = 3x^2 + 2x\).
  • Substitute the given input value into the function. In our case, when \(x = 5\), substitute to get \((f+g)(5) = 3(5)^2 + 2(5)\).
  • Simplify the expression to find the numerical value. This results in \((f+g)(5) = 75 + 10 = 85\).
Function evaluation is a powerful tool for verifying results and exploring mathematical models.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations (addition, subtraction, multiplication, and division) that represent a particular value or range of values.
Key elements of algebraic expressions include:
  • Variables: symbols like \(x\) or \(y\) that represent unknown or changeable values.
  • Constants: fixed numbers such as \(3\) or \(-1\) in the expressions.
  • Coefficients: numbers that multiply the variables, like \(3\) in \(3x^2\).
  • Operators: such as \(+\), \(-\), and \(\times\), which connect the terms together.
In algebra, understanding how to manipulate these elements enables solving equations and exploring mathematical concepts. The exercise we explored demonstrates how to combine and simplify algebraic expressions through function operations.

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Most popular questions from this chapter

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)=\sqrt{x+2}-2 $$

Begin by graphing the standard cubic function, \(f(x)=x^{3} .\) Then use transformations of this graph to graph the given function. $$ h(x)=-x^{3} $$

During a particular year, the taxes owed, \(T(x),\) in dollars, filing separately with an adjusted gross income of \(x\) dollars is given by the piecewise function $$ T(x)=\left\\{\begin{array}{ll} 0.15 x & \text { if } 0 \leq x<17,900 \\ 0.28(x-17,900)+2685 & \text { if } 17,900 \leq x<43,250 \\ 0.31(x-43,250)+9783 & \text { if } x \geq 43,250 \end{array}\right. $$ In Exercises \(89-90,\) use this function to find and interpret each of the following. 90\. \(T(70,000)\)

Describe one advantage of using \(f(x)\) rather than \(y\) in a function's equation. How is the domain of a function determined?

You will be developing functions that model given conditions. A chemist working on a flu vaccine needs to mix a \(10 \%\) sodium-iodine solution with a \(60 \%\) sodium-iodine solution to obtain a 50 -milliliter mixture. Write the amount of sodium iodine in the mixture, \(S,\) in milliliters, as a function of the number of milliliters of the \(10 \%\) solution used. Then find and interpret \(S(30)\)
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