Identify the vertical asymptote and the hole on the graph of the function f(x) = x2+x-6/x2-6x+8.     there is an asymptote at x=          -->   4 there is a whole at                 -->   (2,2.5)

There is an asymptote at x = 4 and 2

and hole at x=2

Explanation:

To find  vertical asymptote  simply equate denominator to zero and find the value of x which is the vertical asymptote.

and hole is the point where numerator and denominator gives equal value

So, to find hole find fractions of numerator and denominator separately the common factor is the hole.

Here after factorisation the given function

Here, (x-2) is the common factor in numerator and denominator is the hole and value of x at denominator after equating to zero is x=2 ,4

The vertical asymptote is x=4. The function has whole at x=2.

Step-by-step explanation:

The given function is

Find factor form of numerator and denominator.

Cancel out the common factors.

Now equate the denominator equal to 0, to find the vertical asymptote.

The vertical asymptote is x=4.

Equate the cancel factor equal to zero to find the whole.

The function has whole at x=2.

At whole, the value of function is undefined but limit of the function exist.

Therefore the limit of the function at x=2 is -2.5.

a) $ 9.50

b) 17

c) y-intercept (0, 12). this is the initial amount of money that joe had prior to playing any game.

d) x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e) slope is -0.5. joe's balance will reduce by 0.5 for every game that he plays.

f)

domain; [0,24]

range; [0,12]

g) x + 2y = 24

step-by-step explanation:

a)

if joe plays 5 games, this will imply that x = 5. substitute x = 5 in the equation and solve for y;

y = -.50x + 12

y = -0.50(5) + 12

y = 9.50

b)

if joe leaves the arcade with $3.50, this will mean that y = 3.50. plug in y = 3.50 in the given equation and solve for x;

y = -.50x + 12

3.50 = -0.50x + 12

0.50x = 12 - 3.50

0.5x = 8.5

x = 17

c)

the function given is   y = -.50x + 12.

the y-intercept is the point where the graph of this function will intersect the y-axis. at this point, the value of x is usually 0. plug in x = 0 in the given equation and determine y;

y = -0.50(0) + 12

y = 12

the y-intercept is thus (0, 12). this is the initial amount of money that joe had prior to playing any game.

d)

the x-intercept refers to the point where the graph of a function intersects with the x-axis. at this point, the value of y is 0. substitute y = 0 in the given equation and solve for x;

0 =   -.50x + 12

0.5x = 12

x = 24

x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e)

the slope of the given equation is -0.5. the equation has been given in slope-intercept form. thus the slope is simply the coefficient of x.

joe's balance will reduce by 0.5 for every game that he plays. this is the interpretation of the slope.

f)

domain is the set of x values for which the function is real and defined. the domain will thus be; [0,24]. this will restrict the balance to only assume positive values.

on the other hand, the range is the set of y values for which a function is real and defined. the range in our case will be [0,12]. joe can have between $0 and $12 balance.

g)

the standard form of the equation of a straight line is;

ax + by = c

where a, b, and c are integers.

our equation is given as;

y = -.50x + 12

re-writing the equation we have;

0.5x + y = 12

divide all through by 0.5;

x + 2y = 24

x=17

step-by-step explanation:

solve 3x-24=x+10

then that makes it 2x=34 because you subtract an x from both sides to get rid of the one on the right, and then you add 24 to both sides to get rid of the 24 on the left

divide by both sides by 2 to get your answer so that you can get your answer