微积分网课代修|预备微积分代写precalculus辅导|MATH113 Finding the Domain of a Function Defined by an Equation

微积分网课代修|预备微积分代写precalculus辅导|MATH113 Finding the Domain of a Function Defined by an Equation

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微积分网课代修|预备微积分代写precalculus辅导|MATH113 Finding the Domain of a Function Defined by an Equation

微积分网课代修|预备微积分代写precalculus辅导|Finding the Domain of a Function Defined by an Equation

In Functions and Function Notation, we were introduced to the concepts of domain and range. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0 .

We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See Figure 2 .

We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [when the set includes the endpoint and a parenthesis (to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $\$ 100$ to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write $(0,100]$. We will discuss interval notation in greater detail later.

微积分网课代修|预备微积分代写precalculus辅导|Finding the Domain of a Function Involving a Denominator

Find the domain of the function $f(x)=\frac{x+1}{2-x}$.
Solution When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for $x$.
$$
\begin{aligned}
2-x &=0 \
-x &=-2 \
x &=2
\end{aligned}
$$
Now, we will exclude 2 from the domain. The answers are all real numbers where $x<2$ or $x>2$. We can use a symbol known as the union, $\cup$, to combine the two sets. In interval notation, we write the solution: $(-\infty, 2) \cup(2, \infty)$.
Figure 4
In interval form, the domain of $f$ is $(-\infty, 2) \cup(2, \infty)$.

微积分网课代修|预备微积分代写precalculus辅导|MATH113 Finding the Domain of a Function Defined by an Equation

微积分网课代修|预备微积分代写precalculus辅导|Finding the Domain of a Function Defined by an Equation


在函数和函数表示法中,我们被介绍了域和范围的概念。在本节中,我们将练习确定特 定函数的域和范围。请记住,在确定领域和范围时,我们需要考虑在现实世界的例子中 什么是物理上可能的或有意义的,例如上面恐怖电影示例中的门票销售和年份。我们还 需要考虑数学上允许的内容。例如,如果域和范围由实数组成,我们不能包含任何导致 我们取负数的偶数根的输入值。或者在以公式表示的函数中,我们不能在域中包含任何 会导致我们除以 0 的输入值。
我们可以将域可视化为包含“功能机器”的“原材料”的“存放区”,而将范围可视化为机器产 品的另一个“存放区”。参见图 2。
我们可以用区间表示法编写域和范围,它使用括号内的值来描述一组数字。在区间表示 法中,我们使用方括号 [当集合包括端点和括号(表示端点不包括在内或区间是无界
的。例如,如果一个人有 $\$ 100$ 要花费,他或她需要表示大于 0 且小于或等于 100 的区 间并写 $(0,100]$. 稍后我们将更详细地讨论区间符号。


微积分网课代修|预备微积分代写precalculus辅导|Finding the Domain of a Function Involving a Denominator


找到函数的域 $f(x)=\frac{x+1}{2-x}$.
解决方案 当存在分母时,我们只想包含不强制分母为零的输入值。因此,我们将分母设 置为 0 并求解 $x$.
$$
2-x=0-x \quad=-2 x=2
$$
现在,我们将从域中排除 2。答案都是实数 $x<2$ 或者 $x>2$. 我们可以使用一个称为 联合的符号, $\cup$, 组合这两组。用区间表示法,我们写出解决方案:
$$
(-\infty, 2) \cup(2, \infty) \text {. }
$$
图 4
在区间形式中,域 $f$ 是 $(-\infty, 2) \cup(2, \infty)$.

微积分网课代修|预备微积分代写precalculus辅导|MATH113 Finding the Domain of a Function Defined by an Equation
微积分网课代修|预备微积分代写precalculus辅导

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