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Chapter 4: Problem 11

Let \(f(x)=-4 x+7\) and \(g(x)=x^{2}+9 x-2 .\) Find the following function values. $$g(-1)$$

Short Answer

Expert verified
The value of the function g(x) when x = -1 is \(-10\). So, \(g(-1) = -10\).

Step by step solution

01

Write down the function definition for g(x)

Given the function definition: \(g(x) = x^2 + 9x - 2\)
02

Substitute x with -1 in the function definition

Now, let's substitute x with -1 in the function definition: \(g(-1) = (-1)^2 + 9(-1) - 2\)
03

Calculate the values

Now, we need to perform the calculations: \[ \begin{aligned} g(-1) &=(-1)^2 + 9(-1) - 2 \\ &= 1 - 9 - 2 \\ &= -8 - 2 \\ &= -10 \end{aligned} \]
04

Final Answer

The value of the function g(x) when x = -1 is \(-10\). So, \(g(-1) = -10\).

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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 a type of algebraic expression consisting of variables and coefficients. They involve operations such as addition, subtraction, and multiplication by a constant.
These functions can have one or multiple terms, which are known as monomials.
The terms in a polynomial are expressed in the form of powers of variables, typically represented as levels of degrees.
  • The degree of a polynomial is the highest power of the variable in the expression.
  • For instance, in the function \(g(x) = x^2 + 9x - 2\), we notice the highest degree is 2, making it a quadratic polynomial.
  • Polynomial functions can be classified as linear, quadratic, cubic, etc., depending on the degree.
Polynomial functions are fundamental because they help in modeling relationships found in real-world scenarios as they allow us to describe patterns and make predictions.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and arithmetic operators such as addition, subtraction, multiplication, and division.
These expressions do not have an equality sign unlike algebraic equations.
Understanding how to manipulate these expressions is key for solving complex problems in math.
  • Variables, often represented by letters like \(x, y,\) or \(z\), stand in for unknown values.
  • Coefficients are numbers placed in front of the variables, indicating multiplication.
  • Constants are standalone numbers within the expression, unaffected by the variables.
In the example provided, the expression \(g(x) = x^2 + 9x - 2\) shows three terms: \(x^2\) (quadratic term), \(9x\) (linear term), and \(-2\) (constant). Each part has specific roles in shaping the function's graph.
Substitution Method
The substitution method involves replacing a variable in an expression with a specified value.
It is a handy technique for evaluating algebraic expressions and functions at particular points.
This process often helps us find the numerical value of the expression or function.
  • Identify the expression or function where substitution is needed.
  • Swap the variable in the expression with the given number.
  • Simplify the resulting expression using order of operations: parentheses, exponents, multiplication, division, addition, and subtraction (PEMDAS/BODMAS).
In the exercise, adopting the substitution method for \(g(x) = x^2 + 9x - 2\) at \(x = -1\), we replace \(x\) with \(-1\) to compute: \[g(-1) = (-1)^2 + 9(-1) - 2 = -10\]This method is effective and straightforward, ensuring accuracy in calculations.

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Most popular questions from this chapter

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{9}{x+4}$$

Graph each function by making a table of values and plotting points. $$g(x)=-\frac{3}{5} x+2$$

Write an equation of the line perpendicular to the given line and containing the given point. Write the answer in slope-intercept form or in standard form, as indicated. \(x+y=9 ;(-5,-5) ;\) slope-intercept form

Write the slope-intercept form of the equation of the line, if possible, given the following information. contains \((-3,-1)\) and \((2,-3)\)

Write an equation of the line parallel to the given line and containing the given point. Write the answer in slope intercept form or in standard form, as indicated. \(18 x-3 y=9 ;(2,-2) ;\) slope-intercept form
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