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Chapter 2: Problem 44

Evaluate the function for the given value of \(x .\) (See Examples \(5-6)\) \(f(x)=x^{2}+3 x \quad g(x)=\frac{1}{x} \quad h(x)=5 \quad k(x)=\sqrt{x+1}\) a. \(k(-2)\) b. \(k(-1)\) c. \(k(0)\) d. \(k(1)\) e. \(k(3)\)

Short Answer

Expert verified
a. not a real number, b. 0, c. 1, d. \(\sqrt{2}\), e. 2

Step by step solution

01

Understand the function

The function given is \( k(x) = \sqrt{x+1} \). This means to find the value of the function for a given \( x \), you need to substitute \( x \) into the function and evaluate \( \sqrt{x+1} \).
02

Evaluate \( k(-2) \)

Substitute \( x = -2 \) into the function: \( k(-2) = \sqrt{-2 + 1} = \sqrt{-1} \). Since \( \sqrt{-1} \) is not a real number (it is an imaginary number, \( i \)), the value of \( k(-2) \) is not a real number.
03

Evaluate \( k(-1) \)

Substitute \( x = -1 \) into the function: \( k(-1) = \sqrt{-1 + 1} = \sqrt{0} = 0 \). So, \( k(-1) = 0 \).
04

Evaluate \( k(0) \)

Substitute \( x = 0 \) into the function: \( k(0) = \sqrt{0 + 1} = \sqrt{1} = 1 \). So, \( k(0) = 1 \).
05

Evaluate \( k(1) \)

Substitute \( x = 1 \) into the function: \( k(1) = \sqrt{1 + 1} = \sqrt{2} \). So, \( k(1) = \sqrt{2} \).
06

Evaluate \( k(3) \)

Substitute \( x = 3 \) into the function: \( k(3) = \sqrt{3 + 1} = \sqrt{4} = 2 \). So, \( k(3) = 2 \).

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Key Concepts

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

Function Evaluation
In math, evaluating a function means finding the output of the function for a given input value. To do this, we use substitution. We replace the variable in the function with the given value and then simplify the expression.

For example, if we have a function defined as:
\( f(x) = x + 3 \)
and we need to evaluate the function for \(x = 2\), we substitute \(2\) into the function:
\[f(2) = 2 + 3 = 5 \]
So, the value of \(f(2)\) is 5. This same approach is used for more complex functions like those involving square roots or other operations.
Square Root Function
The square root function is a type of function where the output is the square root of the input plus or minus some value inside the root. The general form is: \( k(x) = \sqrt{x + 1} \), which means that for any input \(x\), the function outputs the square root of \(x+1\).

When evaluating a square root function:
  • First, solve the expression inside the square root.
  • Next, find the square root of that value.

If the value inside the square root is negative, the result will be an imaginary number because you can't take the square root of a negative number.
For example: \( k(-2) = \sqrt{-2 + 1} = \sqrt{-1} \) results in an imaginary number.
Imaginary Numbers
Imaginary numbers arise when we take the square root of a negative number. The imaginary unit is denoted as \(i\), where \(i = \sqrt{-1} \).

For instance, if we have \( \sqrt{-1} \), it equals \( i \.\) This is important when evaluating functions that have negative values inside a square root.

For example, evaluating \( k(-2) \) of a function \( k(x) = \sqrt{x + 1} \) results in \( \sqrt{-2 + 1} = \sqrt{-1} = i \.\) Therefore, \( k(-2) \) is not a real number but an imaginary number.
Substitution Method
The substitution method is an essential technique used in math to evaluate or simplify expressions. It involves replacing the variable in a function with a given value.

For example, given the function \( f(x) = x + 3 \,\) to find \( f(2) \,\) we substitute \(x\) with 2:
  • First, write down the function: \( f(2) = 2 + 3 \)
  • Next, perform the arithmetic: \( f(2) = 5 \).

This method helps in breaking down problems into simpler steps. It is widely used, especially in functions involving more complex operations like square roots or polynomials.

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Most popular questions from this chapter

Graph the function. $$ s(x)=\left\\{\begin{array}{ll} -x-1 & \text { for } x \leq-1 \\ \sqrt{x+1} & \text { for } x>-1 \end{array}\right. $$

A sales person makes a base salary of \(\$ 2000\) per month. Once he reaches \(\$ 40,000\) in total sales, he earns an additional \(5 \%\) commission on the amount in sales over \(\$ 40,000 .\) Write a piecewise-defined function to model the sales person's total monthly salary \(S(x)\) (in \(\$)\) as a function of the amount in sales \(x\).

Suppose that the average rate of change of a continuous function between any two points to the left of \(x=a\) is negative, and the average rate of change of the function between any two points to the right of \(x=a\) is positive. Does the function have a relative minimum or maximum at \(a\) ?

Use a graphing utility to approximate the relative maxima and relative minima of the function on the standard viewing window. Round to 3 decimal places. $$ f(x)=-0.6 x^{2}+2 x+3 $$

A car traveling \(60 \mathrm{mph}(88 \mathrm{ft} / \mathrm{sec})\) undergoes a constant deceleration until it comes to rest approximately 9.09 sec later. The distance \(d(t)\) (in ft) that the car travels \(t\) seconds after the brakes are applied is given by \(d(t)=-4.84 t^{2}+88 t,\) where \(0 \leq t \leq 9.09 .\) (See Example 5) a. Find the difference quotient \(\frac{d(t+h)-d(t)}{h}\). Use the difference quotient to determine the average rate of speed on the following intervals for \(t\) : b. [0,2]\(\quad(\) Hint \(: t=0\) and \(h=2)\) c. [2,4]\(\quad(\) Hint \(: t=2\) and \(h=2)\) d. [4,6]\(\quad(\) Hint \(: t=4\) and \(h=2)\) e. [6,8]\(\quad(\) Hint \(: t=6\) and \(h=2)\)
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