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Chapter 5: Problem 17

Let \(f(x)=-3 x+1\) and \(g(x)=x^{2}+2 .\) Find each of the following. $$ (f+g)(x) $$

Short Answer

Expert verified
(f+g)(x) = x^2 - 3x + 3

Step by step solution

01

- Understand the Functions

Identify the given functions. The first function is \(f(x) = -3x + 1\) and the second function is \(g(x) = x^2 + 2\).
02

Adding the Functions

To find \((f+g)(x)\), you need to add the two functions together. This means \((f+g)(x) = f(x) + g(x)\).
03

Substitute and Simplify

Substitute the expressions for \(f(x)\) and \(g(x)\) into the equation: \((f+g)(x) = (-3x + 1) + (x^2 + 2)\). Combine like terms to simplify the result.
04

Final Expression

Combine the linear and quadratic terms: \((f+g)(x) = x^2 - 3x + 3\).

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Key Concepts

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

quadratic function
A quadratic function is a type of polynomial function where the highest exponent of the variable (usually x) is 2. The standard form is written as \[ ax^2 + bx + c \] where \(a\), \(b\), and \(c\) are constants. In the original exercise, the quadratic function given is \[ g(x) = x^2 + 2 \]. Here, \(x^2\) represents the quadratic term, while the +2 is a constant term. Quadratic functions usually produce a parabolic graph, which means they have a U-shaped curve. Depending on the sign of \(a\), the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)). In our case, since the coefficient of \(x^2\) is positive, the parabola would open upwards.
linear function
Linear functions are the simplest form of polynomial functions, having a degree of 1. The standard form of a linear function is \[ mx + b \] where \(m\) is the slope and \(b\) is the y-intercept. The function given in the exercise was \[ f(x) = -3x + 1 \]. Here, -3 is the slope and 1 is the y-intercept. Linear functions produce a straight-line graph. The slope (\(m = -3\) in this case) tells us how steep the line is, and the intercept (\(b = 1\)) tells us where the line crosses the y-axis. An important property of linear functions is that they increase or decrease at a constant rate.
combining like terms
Combining like terms is a fundamental skill in algebra that simplifies expressions and equations. Like terms are terms that have the same variables raised to the same power. For instance, \[ -3x \] and \[ x \] are like terms because they both contain the variable \(x\) raised to the first power. To combine like terms, you simply add or subtract their coefficients.
  • In the given exercise, we had \[ (f+g)(x) = (-3x+1) + (x^2 + 2) \].
  • Here, -3x and x^2 are not like terms and cannot be combined, but 1 and 2 are constants and can be added together. This gives us \[ (f+g)(x) = x^2 - 3x + 3 \], which is the final simplified expression.
Combining like terms is crucial for simplifying expressions, solving equations, and finding function values.

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