# 5.2 Power functions and polynomial functions  (Page 2/19)

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To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol $\text{\hspace{0.17em}}\infty \text{\hspace{0.17em}}$ for positive infinity and $\text{\hspace{0.17em}}\mathrm{-\infty }\text{\hspace{0.17em}}$ for negative infinity. When we say that “ $x\text{\hspace{0.17em}}$ approaches infinity,” which can be symbolically written as $\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}$ we are describing a behavior; we are saying that $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is increasing without bound.

With the positive even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches positive or negative infinity, the $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ values increase without bound. In symbolic form, we could write

[link] shows the graphs of $\text{\hspace{0.17em}}f\left(x\right)={x}^{3},\text{\hspace{0.17em}}g\left(x\right)={x}^{5},$ and $\text{\hspace{0.17em}}h\left(x\right)={x}^{7},$ which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.

These examples illustrate that functions of the form $\text{\hspace{0.17em}}f\left(x\right)={x}^{n}\text{\hspace{0.17em}}$ reveal symmetry of one kind or another. First, in [link] we see that even functions of the form even, are symmetric about the $\text{\hspace{0.17em}}y\text{-}$ axis. In [link] we see that odd functions of the form  odd, are symmetric about the origin.

For these odd power functions, as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches negative infinity, $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ decreases without bound. As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches positive infinity, $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ increases without bound. In symbolic form we write

The behavior of the graph of a function as the input values get very small ( $\text{\hspace{0.17em}}x\to -\infty \text{\hspace{0.17em}}$ ) and get very large ( $\text{\hspace{0.17em}}x\to \infty \text{\hspace{0.17em}}$ ) is referred to as the end behavior    of the function. We can use words or symbols to describe end behavior.

[link] shows the end behavior of power functions in the form $\text{\hspace{0.17em}}f\left(x\right)=k{x}^{n}\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is a non-negative integer depending on the power and the constant.

Given a power function $\text{\hspace{0.17em}}f\left(x\right)=k{x}^{n}\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is a non-negative integer, identify the end behavior.

1. Determine whether the power is even or odd.
2. Determine whether the constant is positive or negative.
3. Use [link] to identify the end behavior.

## Identifying the end behavior of a power function

Describe the end behavior of the graph of $\text{\hspace{0.17em}}f\left(x\right)={x}^{8}.$

The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches infinity, the output (value of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ ) increases without bound. We write as $\text{\hspace{0.17em}}x\to \infty ,f\left(x\right)\to \infty .\text{\hspace{0.17em}}$ As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches negative infinity, the output increases without bound. In symbolic form, as We can graphically represent the function as shown in [link] .

## Identifying the end behavior of a power function.

Describe the end behavior of the graph of $\text{\hspace{0.17em}}f\left(x\right)=-{x}^{9}.$

The exponent of the power function is 9 (an odd number). Because the coefficient is $\text{\hspace{0.17em}}–1\text{\hspace{0.17em}}$ (negative), the graph is the reflection about the $\text{\hspace{0.17em}}x\text{-}$ axis of the graph of $\text{\hspace{0.17em}}f\left(x\right)={x}^{9}.\text{\hspace{0.17em}}$ [link] shows that as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches infinity, the output decreases without bound. As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches negative infinity, the output increases without bound. In symbolic form, we would write

Cos45/sec30+cosec30=
Cos 45 = 1/ √ 2 sec 30 = 2/√3 cosec 30 = 2. =1/√2 / 2/√3+2 =1/√2/2+2√3/√3 =1/√2*√3/2+2√3 =√3/√2(2+2√3) =√3/2√2+2√6 --------- (1) =√3 (2√6-2√2)/((2√6)+2√2))(2√6-2√2) =2√3(√6-√2)/(2√6)²-(2√2)² =2√3(√6-√2)/24-8 =2√3(√6-√2)/16 =√18-√16/8 =3√2-√6/8 ----------(2)
exercise 1.2 solution b....isnt it lacking
I dnt get dis work well
what is one-to-one function
what is the procedure in solving quadratic equetion at least 6?
Almighty formula or by factorization...or by graphical analysis
Damian
I need to learn this trigonometry from A level.. can anyone help here?
yes am hia
Miiro
tanh2x =2tanhx/1+tanh^2x
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation
f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb
favour
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
i am in
Cliff
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
helo
Akash
hlo
Akash
Hello
Hudheifa
which of these functions is not uniformly continuous on 0,1