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Chapter 16: Problem 10

Evaluate the function for the given value of \(x\). $$f(x)=2 x^{2}+x-1 ; x=-2$$

Short Answer

Expert verified
The function value for \(x = -2\) is \(f(-2) = 5\).

Step by step solution

01

Identification

Identify the function as \(f(x) = 2x^2 + x - 1\) and the given value of \(x\) as -2.
02

Substitute

Substitute the given value of \(x\) into the function, whereby \(x = -2\) replaces \(x\) in the function, giving \(f(-2) = 2(-2)^2 + (-2) - 1\).
03

Simplify

Simplify the expression by first evaluating the exponent, then perform the multiplication and finally execute the addition and subtraction in order. This gives \(f(-2) = 2(4) - 2 - 1 = 8 - 2 - 1 = 5\).

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Key Concepts

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

Substitution
Substitution is like stepping into someone else's shoes to experience their journey. In mathematics, this involves replacing variables in an expression or function with actual numbers. For example, in the function given by the exercise, you have a variable \( x \) that needs to be evaluated at a specific point, \( x = -2 \).
  • The first thing you do is identify the function. In this case, \( f(x) = 2x^2 + x - 1 \).
  • Then, wherever you see the variable \( x \), substitute \( -2 \) in its place.
This process requires careful substitution to ensure that each instance of \( x \) in the function is replaced accurately, transforming the function \( f(x) \) to \( f(-2) \). By doing this, the abstract equation transforms into something concrete that can be easily solved through calculation.
Algebraic Simplification
Once you've substituted the value into the function, it's time to simplify the expression. Algebraic simplification is the process of breaking down an expression into its simplest form.
  • Start with evaluating any exponents first, according to the order of operations (often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
  • In this case, calculate \((-2)^2\), which equals \(4\).
  • Next, multiply as needed: \(2 \times 4 = 8\).
  • Then add or subtract each term: \(8 - 2 - 1\).
Every step should be done methodically to avoid errors. This ensures that the expression is fully simplified to a value that is easy to understand and interpret, arriving at our final simplified value of \(5\).
Quadratic Functions
Quadratic functions represent a specific type of polynomial function where the highest degree of the variable is two. They have several interesting features.
  • The general form of a quadratic function is \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.
  • In our example, \( a = 2 \), \( b = 1 \), and \( c = -1 \).
  • The graph of a quadratic function forms a curve called a parabola.
Quadratics are a principal part of algebra and are used to model various real-world problems, including the paths of projectiles and areas of rooms. Understanding the process of evaluating these functions helps in determining the value of the function for particular values of \( x \), which is essential in applications of quadratic equations.

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Most popular questions from this chapter

The kinetic energy of a moving body is given by \(E=\frac{1}{2} m v^{2},\) where \(E\) is the kinetic energy, \(m\) is the mass, and \(v\) is the velocity in meters per second. What is the velocity of a moving body whose mass is \(5 \mathrm{kg}\) and whose kinetic energy is 250 newton-meters? (PICTURE NOT COPY)

It took a motorboat 1 h longer to travel 36 mi against the current than it took to travel \(36 \mathrm{mi}\) with the current. The rate of the current was \(3 \mathrm{mph}\). Find the rate of the boat in calm water.

the hypotenuse of a right triangle measures \(\sqrt{13} \mathrm{cm} .\) The length of one leg is \(1 \mathrm{cm}\) shorter than twice the length of the other leg. Find the lengths of the legs of the right triangle.

Explain why the equation \((x-2)^{2}=-4\) does not have a real number solution.

When using the quadratic formula to solve the equation \(2=-16 t^{2}+24 t+4\) for \(t,\) substitute _____ for \(a\) in the quadratic formula, ______ for \(b,\) and ____ for \(c\).
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