### 数学代写|微积分代写Calculus代写|MATH171

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## 数学代写|微积分代写Calculus代写|Analyzing Graphs of Functions

In Section 1.4, you studied functions from an algebraic point of view. In this section, you will study functions from a graphical perspective.

The graph of a function $f$ is the collection of ordered pairs $(x, f(x))$ such that $x$ is in the domain of $f$. As you study this section, remember that
$x=$ the directed distance from the $y$-axis
$y=f(x)=$ the directed distance from the $x$-axis
as shown in Figure 1.52.

Use the graph of the function $f$, shown in Figure 1.53, to find (a) the domain of $f$, (b) the function values $f(-1)$ and $f(2)$, and (c) the range of $f$.
Solution
a. The closed dot at $(-1,1)$ indicates that $x=-1$ is in the domain of $f$, whereas the open dot at $(5,2)$ indicates that $x=5$ is not in the domain. So, the domain of $f$ is all $x$ in the interval $[-1,5)$.
b. Because $(-1,1)$ is a point on the graph of $f$, it follows that $f(-1)=1$. Similarly, because $(2,-3)$ is a point on the graph of $f$, it follows that $f(2)=-3$.
c. Because the graph does not extend below $f(2)=-3$ or above $f(0)=3$, the range of $f$ is the interval $[-3,3]$.
VCHECKPOINT Now try Exercise 1.
The use of dots (open or closed) at the extreme left and right points of a graph indicates that the graph does not extend beyond these points. If no such dots are shown, assume that the graph extends beyond these points.

By the definition of a function, at most one $y$-value corresponds to a given $x$-value. This means that the graph of a function cannot have two or more different points with the same $x$-coordinate, and no two points on the graph of a function can be vertically above or below each other. It follows, then, that a vertical line can intersect the graph of a function at most once. This observation provides a convenient visual test called the Vertical Line Test for functions.
Vertical Line Test for Functions
A set of points in a coordinate plane is the graph of $y$ as a function of $x$ if and only if no vertical line intersects the graph at more than one point.

## 数学代写|微积分代写Calculus代写|Zeros of a Function

If the graph of a function of $x$ has an $x$-intercept at $(a, 0)$, then $a$ is a zero of the function.
Zeros of a Function
The zeros of a function $f$ of $x$ are the $x$-values for which $f(x)=0$.
Example 3 Finding the Zeros of a Function
Find the zeros of each function.
a. $f(x)=3 x^2+x-10$
b. $g(x)=\sqrt{10-x^2}$
c. $h(t)=\frac{2 t-3}{t+5}$
Solution
To find the zeros of a function, set the function equal to zero and solve for the independent variable.
a.
\begin{aligned} & 3 x^2+x-10=0 \quad \text { Set } f(x) \text { equal to } 0 . \ & (3 x-5)(x+2)=0 \ & \text { Factor. } \ & 3 x-5=0 \ & x=\frac{5}{3} \ & \text { Set } 1 \text { st factor equal to } 0 \text {. } \ & x+2=0 \ & x=-2 \ & \text { Set } 2 \text { nd factor equal to } 0 \text {. } \ & \end{aligned}
The zeros of $f$ are $x=\frac{5}{3}$ and $x=-2$. In Figure 1.55, note that the graph of $f$ has $\left(\frac{5}{3}, 0\right)$ and $(-2,0)$ as its $x$-intercepts.
b. $\sqrt{10-x^2}=0$
Set $g(x)$ equal to 0 .
\begin{aligned} 10-x^2 & =0 \ 10 & =x^2 \ \pm \sqrt{10} & =x \end{aligned}
Square each side.
Add $x^2$ to each side.
Extract square roots.
The zeros of $g$ are $x=-\sqrt{10}$ and $x=\sqrt{10}$. In Figure 1.56, note that the graph of $g$ has $(-\sqrt{10}, 0)$ and $(\sqrt{10}, 0)$ as its $x$-intercepts.

# 微积分代考

## 数学代写|微积分代写Calculus代写|Analyzing Graphs of Functions

$x=$到$y$轴的有向距离
$y=f(x)=$到$x$轴的有向距离

a.“$(-1,1)$”表示“$x=-1$”在$f$的域中，“$(5,2)$”表示“$x=5$”不在该域中。所以，$f$的定义域都是$x$在$[-1,5)$区间内。
b.因为$(-1,1)$是$f$图上的一个点，所以可知$f(-1)=1$。同样，因为$(2,-3)$是$f$图上的一个点，所以$f(2)=-3$。
c.由于图在$f(2)=-3$以下或$f(0)=3$以上不扩展，所以$f$的范围为区间$[-3,3]$。

## 数学代写|微积分代写Calculus代写|Zeros of a Function

$x$的函数$f$的零点是$x$ -值，$f(x)=0$。

A. $f(x)=3 x^2+x-10$
B. $g(x)=\sqrt{10-x^2}$
C. $h(t)=\frac{2 t-3}{t+5}$

a。
\begin{aligned} & 3 x^2+x-10=0 \quad \text { Set } f(x) \text { equal to } 0 . \ & (3 x-5)(x+2)=0 \ & \text { Factor. } \ & 3 x-5=0 \ & x=\frac{5}{3} \ & \text { Set } 1 \text { st factor equal to } 0 \text {. } \ & x+2=0 \ & x=-2 \ & \text { Set } 2 \text { nd factor equal to } 0 \text {. } \ & \end{aligned}
$f$的零点分别是$x=\frac{5}{3}$和$x=-2$。在图1.55中，请注意$f$的图形有$\left(\frac{5}{3}, 0\right)$和$(-2,0)$作为其$x$ -截点。
B. $\sqrt{10-x^2}=0$

\begin{aligned} 10-x^2 & =0 \ 10 & =x^2 \ \pm \sqrt{10} & =x \end{aligned}

$g$的零点分别是$x=-\sqrt{10}$和$x=\sqrt{10}$。在图1.56中，请注意$g$的图形有$(-\sqrt{10}, 0)$和$(\sqrt{10}, 0)$作为其$x$ -截点。

## Matlab代写

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