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Chapter 2: Problem 10

Express the function (or rule) in words. $$k(x)=\sqrt{x+2}$$

Short Answer

Expert verified
Add 2 to \(x\), then take the square root.

Step by step solution

01

Identify the Mathematical Function

The function given is \(k(x) = \sqrt{x+2}\). It is a square root function where the expression inside the square root is \(x + 2\).
02

Recognize the Operation

The operation here is the square root function. It takes the input value \(x\) and adds 2 to it before taking the square root of the result.
03

Express in Words

The function \(k(x)\) can be expressed in words as follows: "For any input \(x\), add 2 to \(x\), and then take the square root of the resulting sum."

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Key Concepts

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

Function Notation
Function notation is a way to clearly express mathematical functions. It uses symbols to define rules or operations applied to an input value. In our exercise, the function is defined as \( k(x) = \sqrt{x+2} \). Here, "\( k(x) \)" is the notation used. The "\( k \)" represents the function, and "\( x \)" serves as the variable or input to the function.
  • "\( k(x) \)" is read as "\( k \) of \( x \)." This indicates that \( x \) is the input value for the function \( k \).
  • The equation part, "\( \sqrt{x+2} \)", displays the operations involved.
Function notation helps simplify complex mathematical concepts by replacing lengthy explanations with concise equations. This allows us to ascertain what kind of manipulation is applied to the input to get the desired output.
Mathematical Operations
Mathematical operations are actions or procedures that produce a new value from one or more inputs. In the context of our function \( k(x) = \sqrt{x+2} \), two main operations occur: addition and square root extraction.
  • First, the input \( x \) is increased by 2, as indicated by \( x + 2 \).
  • Then, the square root of the computed sum (\( x + 2 \)) is taken to produce the final output. This is depicted by the \( \sqrt{} \) symbol.
These steps transform the initial input into the final result through a sequence of calculations. Understanding these operations is essential to grasp how inputs are modified to generate outputs in a function.
Function Expression in Words
Converting mathematical expressions into words can help deepen our understanding of what a function does. The function \( k(x) = \sqrt{x+2} \) can be expressed in simple words as follows:
  • Take an input value \( x \).
  • Add 2 to this value.
  • Calculate the square root of the result obtained from adding 2.
In essence, this verbal expression describes the sequence of operations within the function. By understanding this description, students can better visualize and comprehend the processes occurring in mathematical functions. Writing it out in words makes the function's operations clear and understandable.

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Most popular questions from this chapter

When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. (a) Suppose $$g(x)=\sqrt{f(x)}$$ where \(f(x) \geq 0\) for all \(x .\) Explain why the local minima and maxima of \(f\) and \(g\) occur at the same values of \(x .\) (b) Let \(g(x)\) be the distance between the point \((3,0)\) and the point \(\left(x, x^{2}\right)\) on the graph of the parabola \(y=x^{2} .\) Express \(g\) as a function of \(x\) (c) Find the minimum value of the function \(g\) that you found in part (b). Use the principle described in part (a) to simplify your work.

If the function \(f\) has the same average rate of change \(c\) between any two points, then for the points \(a\) and \(x\) we have $$c=\frac{f(x)-f(a)}{x-a}$$ Rearrange this expression to show that $$f(x)=c x+(f(a)-c a)$$ and conclude that \(f\) is a linear function.

Find the domain of the function. $$f(x)=\frac{x^{2}}{\sqrt{6-x}}$$

Find the domain of the function. $$g(x)=\frac{\sqrt{2+x}}{3-x}$$

Evaluate the function at the indicated values. $$\begin{array}{l} f(x)=2 x+1 ; \\ f(1), f(-2), f\left(\frac{1}{2}\right), f(a), f(-a), f(a+b) \end{array}$$
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