4.1 Linear functions  (Page 2/27)

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$D\left(t\right)=83t+250$

Representing a linear function in tabular form

A third method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in [link] . From the table, we can see that the distance changes by 83 meters for every 1 second increase in time.

Can the input in the previous example be any real number?

No. The input represents time so while nonnegative rational and irrational numbers are possible, negative real numbers are not possible for this example. The input consists of non-negative real numbers.

Representing a linear function in graphical form

Another way to represent linear functions is visually, using a graph. We can use the function relationship from above, $\text{\hspace{0.17em}}D\left(t\right)=83t+250,\text{\hspace{0.17em}}$ to draw a graph as represented in [link] . Notice the graph is a line . When we plot a linear function, the graph is always a line.

The rate of change, which is constant, determines the slant, or slope    of the line. The point at which the input value is zero is the vertical intercept, or y -intercept    , of the line. We can see from the graph that the y -intercept in the train example we just saw is $\text{\hspace{0.17em}}\left(0,250\right)\text{\hspace{0.17em}}$ and represents the distance of the train from the station when it began moving at a constant speed.

Notice that the graph of the train example is restricted, but this is not always the case. Consider the graph of the line $\text{\hspace{0.17em}}f\left(x\right)=2x+1.\text{\hspace{0.17em}}$ Ask yourself what numbers can be input to the function. In other words, what is the domain of the function? The domain is comprised of all real numbers because any number may be doubled, and then have one added to the product.

Linear function

A linear function    is a function whose graph is a line. Linear functions can be written in the slope-intercept form    of a line

$f\left(x\right)=mx+b$

where $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is the initial or starting value of the function (when input, $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ ), and $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ is the constant rate of change, or slope of the function. The y -intercept is at $\text{\hspace{0.17em}}\left(0,b\right).$

Using a linear function to find the pressure on a diver

The pressure, $\text{\hspace{0.17em}}P,$ in pounds per square inch (PSI) on the diver in [link] depends upon her depth below the water surface, $\text{\hspace{0.17em}}d,$ in feet. This relationship may be modeled by the equation, $\text{\hspace{0.17em}}P\left(d\right)=0.434d+14.696.\text{\hspace{0.17em}}$ Restate this function in words.

To restate the function in words, we need to describe each part of the equation. The pressure as a function of depth equals four hundred thirty-four thousandths times depth plus fourteen and six hundred ninety-six thousandths.

Determining whether a linear function is increasing, decreasing, or constant

The linear functions we used in the two previous examples increased over time, but not every linear function does. A linear function may be increasing, decreasing, or constant. For an increasing function    , as with the train example, the output values increase as the input values increase. The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right as in [link] (a) . For a decreasing function    , the slope is negative. The output values decrease as the input values increase. A line with a negative slope slants downward from left to right as in [link] (b) . If the function is constant, the output values are the same for all input values so the slope is zero. A line with a slope of zero is horizontal as in [link] (c) .

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
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
SO THE ANSWER IS X=-8
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
1KI POWER 1/3 PLEASE SOLUTIONS
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
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin