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Chapter 5: Problem 66

Find the specified function values. Find \(Q(3)\) and \(Q(-1): Q(x)=-4 x^{3}+7 x^{2}-6\)

Short Answer

Expert verified
Q(3) = -51 and Q(-1) = 5.

Step by step solution

01

- Understand the Given Function

Identify the given function. Here, the function is given as Q(x) = -4x^3 + 7x^2 - 6. Our task is to find the values of Q(3) and Q(-1).
02

- Substitute x with 3 to Find Q(3)

Replace x with 3 in the function Q(x). This gives us: Q(3) = -4(3)^3 + 7(3)^2 - 6. Now, calculate each term: Q(3) = -4(27) + 7(9) - 6 Q(3) = -108 + 63 - 6.
03

- Simplify Q(3)

Add the previous results to find Q(3): Q(3) = -108 + 63 - 6 Q(3) = -108 + 57 Q(3) = -51.
04

- Substitute x with -1 to Find Q(-1)

Replace x with -1 in the function Q(x). This gives us: Q(-1) = -4(-1)^3 + 7(-1)^2 - 6. Now, calculate each term: Q(-1) = -4(-1) + 7(1) - 6 Q(-1) = 4 + 7 - 6.
05

- Simplify Q(-1)

Add the previous results to find Q(-1): Q(-1) = 4 + 7 - 6 Q(-1) = 5.

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Key Concepts

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

Function Substitution
Function substitution means replacing the variable in a function with a given value. This helps us find the function's output for specific inputs. For instance, if we have a function Q(x)=-4x^3 + 7x^2 - 6, and we need to find Q(3), we substitute x with 3.
Substitution changes the function to: Q(3) = -4(3)^3 + 7(3)^2 - 6.
Remember to carefully follow the order of operations:
  • Calculate the exponents
  • Then multiply
  • Finally, add or subtract the results.
    • This method allows precise solutions specific to the input values.
    Polynomials
    Polynomials are a type of algebraic expression that include variables raised to whole number powers and their coefficients. In the function Q(x)=-4x^3 + 7x^2 - 6,
    Q(x) is a cubic polynomial because the highest power of x is 3.
    Here are the components of our polynomial:
  • The term -4x^3 is the cubic term.
  • The term 7x^2 is the quadratic term.
  • The term -6 is the constant term.
    • Polynomials are vital in algebra because they can represent complex relationships and have multiple uses in calculus and real-world problems.
    Algebraic Expressions
    Algebraic expressions combine variables, coefficients, and constants using arithmetic operations. In our exercise, Q(x)=-4x^3 + 7x^2 - 6 is an algebraic expression.
    Breaking it down:
  • -4x^3 represents 'minus four times x cubed,'
  • 7x^2 means 'seven times x squared,'
  • -6 is the constant.
    • When working with algebraic expressions:
  • Identify each term and its components.
  • Observe how terms are connected using + or - signs.
    • Simplifying algebraic expressions requires correctly following arithmetic rules and operations.
    Step-by-Step Solution
    Solving problems step-by-step ensures that we don't miss crucial parts and helps us understand each process. Here's a simplified approach for our exercise.
    First, we copy the given polynomial: Q(x) = -4x^3 + 7x^2 - 6.
    Next, for Q(3), we substitute x with 3: Q(3) = -4(3)^3 + 7(3)^2 - 6.
    Evaluate each component: -4(27) + 7(9) - 6 = -108 + 63 - 6
    Finally, add the results to get Q(3) = -51.
    Repeat these steps for Q(-1): Q(-1) = -4(-1)^3 + 7(-1)^2 - 6
    Evaluate the terms: -4(-1) + 7(1) - 6 = 4 + 7 - 6
    Add the results to find Q(-1) = 5.
    This step-by-step method helps you approach any similar problems systematically.

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    Most popular questions from this chapter

    F(x)\( and \)g(x)\( are as given. Find a simplified expression for \)F(x)\( if \)F(x)=(f / g)(x)$. $$ f(x)=64 x^{3}-8, g(x)=4 x-2 $$

    For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=x+\frac{2}{x-1}\\\ &g(x)=3 x^{3} \end{aligned} $$

    Review factoring expressions and solving equations. $$ 3 x-1=0 $$

    Use the fact that \(10^{3} \approx 2^{10}\) to estimate each of the following powers of \(2 .\) Then compute the power of 2 with a calculator and find the difference between the exact value and the approximation. $$ 2^{26} $$

    For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=2 x^{3}\\\ &g(x)=5-x \end{aligned} $$
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