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Chapter 3: Problem 29

State the instructions of the function in words as in Example \(9 .\) $$f(x)=x^{2}+2$$

Short Answer

Expert verified
Square the input and add 2 to get the output.

Step by step solution

01

Identify the Function Type

Analyze the given function to determine its type. The function is provided as \( f(x) = x^2 + 2 \). This is a quadratic function as it includes an \( x^2 \) term.
02

Recognize the Standard Form

Rewrite the function in the standard form for quadratic functions: \( f(x) = ax^2 + bx + c \). Here, \( a = 1 \), \( b = 0 \), and \( c = 2 \).
03

Describe the Instructions in Words

To describe the function \( f(x) = x^2 + 2 \) in words, follow these instructions: * Take any input value \( x \). * Square the input value (multiply \( x \) by itself to get \( x^2 \)). * Add 2 to the result from the previous step to find the output: \( x^2 + 2 \).

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Key Concepts

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

Function Analysis
Function analysis is the process of understanding the characteristics and behavior of a function. Here, we review a specific function, \( f(x) = x^2 + 2 \). The primary step in function analysis is to identify the type of function.

The given function is a quadratic function. This is determined by the presence of the \( x^2 \) term鈥攎aking it a second-degree polynomial. Quadratic functions often have a characteristic U-shaped curve known as a parabola.

When conducting function analysis, consider these attributes:
  • The highest exponent (degree) of the variable \( x \).
  • The number of terms, as quadratic functions usually comprise three: \( ax^2 \), \( bx \), and \( c \).
  • Graph shape and direction, where the parabola opens upwards if \( a > 0 \), or downwards if \( a < 0 \).
Standard Form of Quadratic Functions
The standard form of a quadratic function is essential for understanding and manipulating quadratic expressions. It is denoted as \( f(x) = ax^2 + bx + c \). This form helps in analyzing the properties and behavior of the quadratic equation easily.

In the given function \( f(x) = x^2 + 2 \), it can be expressed in standard form as \( f(x) = 1x^2 + 0x + 2 \) where:
  • \( a = 1 \), the coefficient of \( x^2 \).
  • \( b = 0 \), the coefficient of \( x \). It indicates that there is no linear term present.
  • \( c = 2 \), the constant term.
Understanding the standard form helps to easily graph the function, determine the vertex, and find roots or solutions.
Describing Functions in Words
Describing a function in words involves explaining how to process an input value using the rules of the function to yield an output. This reinforces understanding and makes it accessible even when not working with numbers or graphs.

Let's describe \( f(x) = x^2 + 2 \) in words:
  • Start by taking an input value, which is any real number \( x \).
  • Square this input value. This means you multiply \( x \) by itself to get \( x^2 \).
  • Add 2 to the result from the squaring step. This sum is your final output, represented as \( x^2 + 2 \).
Explaining functions this way aids in comprehension, especially for visual and auditory learning methods, providing a clear step-by-step process of the function's operations.

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Most popular questions from this chapter

Solve the given problems. A hot-water faucet is turned on. (a) Sketch a reasonable graph of the water temperature as a function of time. (b) Compare to a typical situation described by \(T=\frac{t^{3}+80}{0.015 t^{3}+4},\) where \(T\) is the water temperature (in \(^{\circ} \mathrm{C}\) ) and \(t\) is the time (in \(\mathrm{s}\) ).

Graph the indicated functions. The resistance \(R\) (in \(\Omega\) ) of a resistor as a function of the temperature \(T\) (in \(^{\circ} \mathrm{C}\) ) is given by \(R=250(1+0.0032 T) .\) Plot \(R\) as a function of \(T\)

Solve the given problems. Display the graph of \(y=c x^{3}\) with \(c=-2\) and with \(c=2\) Describe the effect of the value of \(c .\)

Use the following table, which gives the fraction (as a decimal) of the total heating load of a certain system that will be supplied by a solar collector of area \(A\) (in \(\mathrm{m}^{2}\) ). Find the indicated values by linear interpolation. $$\begin{array}{l|c|c|c|c|c|c|c} f & 0.22 & 0.30 & 0.37 & 0.44 & 0.50 & 0.56 & 0.61 \\ \hline A\left(\mathrm{m}^{2}\right) & 20 & 30 & 40 & 50 & 60 & 70 & 80 \end{array}$$ For \(A=52 \mathrm{m}^{2},\) find \(f.\)

Answer the given questions. Join the points (-1,-2),(4,-2),(7,2),(2,2) and (-1,-2) in order with straight line segments. Find the distances between successive points and then identify the geometric figure formed.
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