/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 46 Find \(f(1), f(-2),\) and \(f(3)... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 46

Find \(f(1), f(-2),\) and \(f(3)\) $$f(x)=x^{2}-2$$

Short Answer

Expert verified
Therefore, \(f(1) = -1, f(-2) = 2\), and \(f(3) = 7\).

Step by step solution

01

Substitute x = 1

Plug x = 1 into the equation: \(f(1) = (1)^{2}-2 \). Then, perform the arithmetic operation to find the value.
02

Substitute x = -2

Plug x = -2 into the equation: \(f(-2) = (-2)^{2}-2 \). Then, perform the arithmetic operation to find the value.
03

Substitute x = 3

Plug x = 3 into the equation: \(f(3) = (3)^{2}-2 \). Then, perform the arithmetic operation to find the value.

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Key Concepts

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

Substitution
Substitution is a fundamental technique used in mathematics to evaluate functions at specific points. In this process, we simply replace the variable, often represented by "x," with a given number in the function's equation. This allows us to calculate the function's output for that particular input.

To substitute, follow these steps:
  • Identify the function. In our case, it is \(f(x) = x^2 - 2\).
  • Take the given values of \(x\), such as 1, -2, and 3.
  • Replace \(x\) with these numbers one at a time in the equation.
This transforms the function into a simple arithmetic problem, where you follow the order of operations to compute the result. Substitution helps in understanding how changes in input affect the output, a crucial concept in calculus and algebra.
Quadratic Function
A quadratic function is defined as a polynomial function of degree two, typically in the form of \(f(x) = ax^2 + bx + c\). In our exercise, the quadratic function is \(f(x) = x^2 - 2\), which is a simplified version where \(a = 1\), \(b = 0\), and \(c = -2\).

Key characteristics of quadratic functions include:
  • Parabola Shape: Graphs of quadratic functions create a U-shaped curve called a parabola. This graph opens upwards if the coefficient of the \(x^2\) term is positive and downwards if it is negative.
  • Vertex: This is the highest or lowest point on the parabola, depending on its orientation.
  • Axis of Symmetry: A vertical line through the vertex, dividing the parabola into two symmetrical halves.
Understanding these properties helps in identifying the nature of the solutions and the type of transformations that occur when adjusting parameters like \(a\), \(b\), and \(c\).
Arithmetic Operations
Arithmetic operations are the foundation of evaluating expressions and solving equations in mathematics. They include basic operations such as addition, subtraction, multiplication, and division. For quadratic functions, like \(f(x) = x^2 - 2\), these operations are used after substitution to find the exact values of the function at different points.

Here's how they apply in our example:
  • Substitution: First, we plug in the values of \(x\), resulting in expressions like \((1)^2 - 2\).
  • Exponents: Apply the power to the variable or number, calculating \(x^2\).
  • Subtraction: Complete the operation by subtracting 2 from \(x^2\) to find the function's value.
Completing these operations correctly ensures that you obtain accurate results. These skills are crucial not only for mathematics but also in various applications involving data analysis, computer science, and scientific computation.

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Most popular questions from this chapter

If points \(P(a, b)\) and \(Q(c, d)\) are two points on a rectangular coordinate system and point \(M\) is midway between them, then point \(M\) is called the midpoint of the line segment joining \(P\) and \(Q .\) (See the illustration on the following page. To find the coordinates of the midpoint \(M\left(x_{M}, y_{M}\right)\) of the segment PQ, we find the average of the \(x\) -coordinates and the average of the \(y\)-coordinates of \(P\) and \(Q\). $$x_{M}=\frac{a+c}{2}$$ and $$y_{M}=\frac{b+d}{2}$$ Find the coordinates of the midpoint of the line segment with the given endpoints. $$P(-8,12) \text { and } Q(3,-9)$$

Fill in the blanks. Assume that \(k\) is a constant. The equation \(y=k x\) represents _____ variation.

Set up a variation equation and solve for the requested value. The distance that a car can travel without refueling varies directly with the number of gallons of gasoline in the tank. If a car can go 360 miles on 12 gallons of gas, how far can it go on 7 gallons?

Set up a variation equation and solve for the requested value. For a fixed area, the length of a rectangle is inversely proportional to its width. A rectangle has a width of 8 feet and a length of 10 feet. If the length is increased to 16 feet, find the width of the rectangle.

Express each variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. \(c\) varies jointly with \(d\) and \(h .\) If \(c=196\) when \(d=6\) and \(h=4,\) find \(d\) when \(c=705.6\) and \(h=16\).
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