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hoa [83]
3 years ago
8

Write 6.9 as a mixed number in simplest form.

Mathematics
1 answer:
BartSMP [9]3 years ago
6 0
Ithw anawer is 6 9/10
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Firlakuza [10]

Answer:

  • Domain: All the real values except x = 2 and x = 4: R - {2, 4}
  • Holes: x = 2
  • VA, vertical asymptores: x = 4
  • HA: horizontal asymptotes: there are not horizontal asymptotes
  • OA: oblique asymptotes: x + 6 [note that OH does not stand for any known feature, and so it is understood that it was intended to write OA]
  • Roots: x = 2
  • Y-intercept: -1

Step-by-step explanation:

1. <u>Given</u>:

f(x)=\frac{x^3-8}{x^2-6x+8}

  • Note that the number 8 in the numerator is not part of the power.
  • Type of function: rational function

2. <u>Domain</u>: is the set of x-values for which the function is defined.

The given function is defined for all x except those for which the denominator equals 0.

  • Denominator:  x² -6x + 8 = 0
  • Solve for x:

        Factor. (x - 4 )(x - 2) = 0

        Zero product property: (x - 4) = 0 or (x - 2) = 0

        x - 4 = 0 ⇒ x = 4

        x - 2 = 0 ⇒ x = 2

  • Domain:

        All the real values except x = 2 and x = 4: x ∈ R / x ≠ 2 and x ≠ 4.

3. <u>Holes</u>:

The holes on the graph of a rational function are at those x-values for which both the numerator and denominator are zero.

  • Find the values for which the numerator is zero:

        Numerator: x³ - 8 = 0

        Factor using difference of cubes property:

                   a³ - b³ = (a - b)(a² + ab + b²)

                   x³ - 8 = (x - 2)(x² + 2x + 4) = 0

        Zero product property:  (x - 2)(x² + 2x + 4) = 0

                    x - 2 = 0 ⇒ x = 2                    

                    x² + 2x + 4 = 0 (this has not real solution)

  • The values for which the denominator is zero were determined above: x = 2 and x = 4.

  • Conclusion: for x = 2 both numerator and denominator equal 0, so this is a hole.

4. <u>VA: Vertical asymptotes</u>.

The vertical asymptotes on the graph of a rational function are the vertical lines for which only the denominator (and not the numerator) equals zero.

  • In the previous part it was determined that happens when x = 4.

5. <u>HA: Horizontal asymptotes</u>.

In rational functions, if the numerator is a higher degree polynomial than the denominator, there is no horizontal asymptote.

6. <u>OA: oblique asymptotes</u>

  • Find the quotient and the remainder.

                       x + 6

                  _______________

x² - 6x + 8 )   x³ + 0x² + 0x - 8

                  - x³ + 6x² - 8x

                   ___________

                          6 x² -   8x -  8

                        - 6x² + 36x - 48

                        _____________

                                    28x  - 56

Result: (x + 6) + (28x - 56) / (x² - 6x + 8)

  • Find limit x → ∞

\lim_{x \to \infty}(x + 6) + \frac{28x-56}{x^2-6x+8}=x+6

<u>7. Roots</u>:

Roots are the values for which f(x) = 0.

That happens when the numerator equals 0, and the denominator is not 0.

As determined earlier: x³ - 8 = 0 ⇒ x = 2.

8. <u>Y-Intercept</u>

The y-intercepts of any function are the y-values when x = 0

  • Substitute x = 0 into the function:

         f(x)=\frac{x^3-8}{x^2-6x+8}=\frac{0^3-8}{0^2-6(0)+8}}=\frac{-8}{8} =-1

3 0
3 years ago
Susan and terry run a day care center Since they use susans house it was agreed that her share is twice as Terrys if they earn 2
Nadya [2.5K]
The easiest way to do this is to divide 225 by 3, so 225 / 3, and what you get as the sum, in this case it's 75, you would multiply the sum by 2, which equals 150, and that, would be Susan's share. So... 225 / 3 = 75. 75 * 2 = 150. $150 is the amount Susan gets. Hope this helps!
6 0
3 years ago
Consider the following system of equations:
Mashutka [201]

Answer:

Liney=-2x+3 intersects line y=x-5

Step-by-step explanation:

We are given that

y=-2x+3

y=x-5

Subtract one equation from other then we get

-3x+8=0

3x=8

x=\frac{8}{3}

Substitute the value of x in first equation then we get

y=-2(\frac{8}{3})+3=-\frac{16}{3}+3=-\frac{-7}{3}

Hence, the solution (\frac{8}{3},-\frac{7}{3}) is the intersection point of two line  equations .

Answer:Liney=-2x+3 intersects line y=x-5

6 0
3 years ago
A circle with a radius of 3 has a central angle of 17/9 pi what is the area of the sector
mafiozo [28]

Answer:17/2pi

Explained:

Just writing longer so I don’t have to go into details. If this is on khan academy you don’t have to show your work!

3 0
3 years ago
Find the perimeter of the shaded region. Round your answer to the nearest hundredth. Help me por favor
timurjin [86]

Answer:

19.15

Step-by-step explanation:

so find the circumference of the circle using c=2*pi*r it's like 18.849. then find the perimeter of the box so p=2l+2w and it's like 38. so subtract the circle circumference from 38 so 38-18.849=19.15

6 0
3 years ago
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