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Chapter 3: Problem 35

Let \(g(x)=2,\) for all \(x\). Find each output. (a) \(g(0)\) (b) \(g(5)\) (c) \(g(x+h)\)

Short Answer

Expert verified
Each output is 2, regardless of the input.

Step by step solution

01

Understanding the function

The function is defined as a constant function where \( g(x) = 2 \) for all \( x \). This means regardless of the input value, the output will always be \( 2 \).
02

Evaluate g(0)

Since \( g(x) = 2 \) for all \( x \), it follows that \( g(0) = 2 \).
03

Evaluate g(5)

Similarly, using the constant function definition, \( g(5) = 2 \).
04

Evaluate g(x+h)

For \( g(x+h) \), since the function outputs \( 2 \) for any input, \( g(x+h) = 2 \) as well.

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Key Concepts

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

Function Evaluation
Evaluation of a function involves finding the output value when a particular input is substituted into the function. In the context of a constant function like \( g(x) = 2 \), function evaluation is straightforward. No matter the input value \( x \), the output is always the constant value, which is 2 in this case.

To understand this concept thoroughly:
  • Identify the function type. Here, it is a constant function.
  • Recognize that for every input \( x \), the output will remain the same.
  • Apply this understanding to any given input, such as a number or an expression involving variables.
For example, in the problem, evaluating \( g(0) \), \( g(5) \), and \( g(x+h) \) all results in 2, since the function does not change with different inputs.
Precalculus Concepts
In precalculus, understanding the basics of functions and their types is crucial. Constant functions are one of the simplest forms of functions you will encounter. A constant function has the form \( f(x) = c \), where \( c \) is a constant. This simplicity makes them an excellent starting point for grasping more complex ideas later on.

Key points about constant functions include:
  • They are horizontal lines on a graph which means they graph as a straight line parallel to the x-axis.
  • They do not vary with changes in \( x \), so the rate of change (slope) is zero.
  • They serve as a foundation for understanding more complicated functions where outputs depend greatly on inputs.
By understanding these basic properties, students can better comprehend the role constant functions play in the broader mathematical landscape of precalculus.
Constant Value
A constant value in mathematics is a fixed value that does not change irrespective of the inputs or operations performed on it. In the case of the function \( g(x) = 2 \), the constant value is 2.

Characteristics of constant values include:
  • Predictability: They ensure the output remains unchanged despite any input variation.
  • Simplicity: Calculations involve less complexity since the result for any input is known.
  • Graphical Representation: In graphs, constant values form horizontal lines at the constant value level.
This concept teaches students the importance of recognizing when a value in a function remains constant, which is paramount in both simplifying calculations and understanding the behavior of different types of functions.

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Most popular questions from this chapter

Let \(P\) be a point with coordinates \((a, b),\) and assume that \(c\) and \(d\) are positive numbers. (The condition that \(c\) and \(d\) are positive isn't really necessary in this problem, but it will help you to visualize things.) (a) Translate the point \(P\) by \(c\) units in the \(x\) -direction to obtain a point \(Q,\) then translate \(Q\) by \(d\) units in the y-direction to obtain a point \(R\). What are the coordinates of the point \(R ?\) (b) Translate the point \(P\) by \(d\) units in the \(y\) -direction to obtain a point \(S,\) then translate \(S\) by \(c\) units in the \(x\) -direction to obtain a point \(T .\) What are the coordinates of the point \(T ?\) (c) Compare your answers for parts (a) and (b). What have you demonstrated? (Answer in complete sentences.)

Use the given function and compute the first six iterates of each initial input \(x_{0}\). In cases in which a calculator answer contains four or more decimal places, round the final answer to three decimal places. (However, during the calculations, work with all of the decimal places that your calculator affords.) \(g(x)=\frac{1}{4} x+3\) (a) \(x_{0}=3\) (b) \(x_{0}=4\) (c) \(x_{0}=5\)

Use this definition: A prime number is a positive whole number with no factors other than itself and \(1 .\) For example, \(2,13,\) and 37 are primes, but 24 and 39 are not. \(B y\) convention 1 is not considered prime, so the list of the first few primes is as follows: \(2,3,5,7,11,13,17,19,23,29, \ldots\) (a) If \(P(x)=x^{2}-x+17,\) find \(P(1), P(2), P(3),\) and \(P(4)\) Can you find a natural number \(x\) for which \(P(x)\) is not prime? (b) If \(Q(x)=x^{2}-x+41,\) find \(Q(1), Q(2), Q(3),\) and \(Q(4)\) Can you find a natural number \(x\) for which \(Q(x)\) is not prime?

Use the given function and compute the first six iterates of each initial input \(x_{0}\). In cases in which a calculator answer contains four or more decimal places, round the final answer to three decimal places. (However, during the calculations, work with all of the decimal places that your calculator affords.) \(F(x)=x^{2}\) (a) \(x_{0}=0.9\) (b) \(x_{0}=1\) (c) \(x_{0}=1.1\)

Use the horizontal line test to determine whether the function is one-to-one (and therefore has an inverse ). (You should be able to sketch the graph of each function on your own, without using a graphing utility.) $$g(x)=\left\\{\begin{array}{ll}x^{2} & \text { if }-1 \leq x<0 \\\x^{2}+1 & \text { if } x \geq 0\end{array}\right.$$
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