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Chapter 2: Problem 51

Evaluate the function for the given value of \(x .\) (See Examples \(5-6)\) \(f(x)=x^{2}+3 x \quad g(x)=\frac{1}{x} \quad h(x)=5 \quad k(x)=\sqrt{x+1}\) $$k(8)$$

Short Answer

Expert verified
饾憳(8)=3

Step by step solution

01

Identify the function

Determine which function needs to be evaluated. In this case, it is the function given by .
02

Plug in the Value of 饾懃

Substitute the given value 饾懃=8 into the function 饾憳(饾懃). The function 饾憳(饾懃) is: 鈭� 鈭氿潙�+1鈭� Thus, 饾憳(8)=鈭�8+1.
03

Perform the Calculation

Calculate the expression 鈭� 鈭�8+1鈭� , which results in 鈭�9 .
04

Solve the Expression

Simplify 鈭�9 to get the final answer . Since 鈭�9=3,

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Key Concepts

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

function evaluation
Function evaluation is a key concept in algebra where you determine the output of a function for a given input value.
Functions act like machines that take an input, process it, and give an output.
In our example, we are given a function, specifically 饾憳(饾懃)=鈭�(饾懃+1).
To evaluate 饾憳(8), you simply substitute the value 8 for 饾懃 in the function to find 饾憳(8).
substitution
Substitution is a crucial step in the process of function evaluation.
This involves replacing the variable (commonly 饾懃) in the function with the provided value.
In the given problem, you substitute 饾懃 with 8 in the function 饾憳(饾懃).
So we start with the function 饾憳(饾懃)=鈭�(饾懃+1). Substituting 饾懃 with 8, we get:
  • 饾憳(8)=鈭�(8+1).

This substitution helps to personalize the function and simplifies the evaluation process.
square root calculation
Once substituted, the next step is to perform the arithmetic calculation required, which often involves square root calculations.
Square root calculations are used to find a number which, when multiplied by itself, yields the original number.
In this example, we need to calculate 鈭�(8+1), so we first add 8 and 1 inside the square root:
  • 鈭�(8+1) = 鈭�9.

Knowing the square root of 9 helps simplify the expression efficiently.
simplification
The final and often easiest step in evaluating a function is the simplification.
Simplification means reducing the expression to its simplest form.
For our problem, after substituting and calculating the square root, we ended up with 鈭�9.
The square root of 9 is 3, as 3脳3=9.
Therefore, the simplified output for the function k(8) is 3.
Simplification helps to make the solution clearer and more understandable.
Let's recap the entire process:
  • Start with 饾憳(饾懃)=鈭�(饾懃+1).
  • Substitute 饾懃 with 8: 饾憳(8)=鈭�(8+1).
  • Calculate the square root: 鈭�(8+1)=鈭�9.
  • Simplify the result: 鈭�9=3.

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Most popular questions from this chapter

The number of cigarettes, \(N(x)\) (in billions), produced in the United States can be approximated by \(N(x)=-0.28 x^{2}+\) \(20.8 x+194\) where \(x\) is the number of years since \(1940 .\) (Source: Centers for Disease Control, www.cdc.gov) a. Determine the difference quotient. \(\frac{N(x+h)-N(x)}{h}\) b. Evaluate the difference quotient on the interval \([10,20],\) and interpret its meaning in the context of this problem. c. Evaluate the difference quotient on the interval \([50,60],\) and interpret its meaning in the context of this problem.

Use a graphing utility to graph the piecewise-defined function. $$ \begin{aligned} &k(x)=\left\\{\begin{array}{ll} -2.7 x-4.1 & \text { for } x \leq-1 \\ -x^{3}+2 x+5 & \text { for }-1

Evaluate the function for the given values of \(x\). \(f(x)=\left\\{\begin{aligned}-3 x+7 & \text { for } x<-1 \\ x^{2}+3 & \text { for }-1 \leq x<4 \\ 5 & \text { for } x \geq 4 \end{aligned}\right.\) a. \(f(3)\) b. \(f(-2)\) c. \(f(-1)\) d. \(f(4)\) e. \(f(5)\)

A car starts from rest and accelerates to a speed of \(60 \mathrm{mph}\) in \(12 \mathrm{sec} .\) It travels \(60 \mathrm{mph}\) for \(1 \mathrm{~min}\) and then decelerates for 20 sec until it comes to rest. The speed of the car \(s(t)\) (in mph) at a time \(t\) (in sec) after the car begins motion can be modeled by: \(s(t)=\left\\{\begin{array}{cl}\frac{5}{12} t^{2} & \text { for } 0 \leq t \leq 12 \\ 60 & \text { for } 12

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