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1 year ago

What is the domain of the function yx4

Mathematics

1 answer:

lutik1710 [3]1 year ago

5 0

The domain of the function is [-4,∞).

Domain is the set of all the possible inputs for which the function is defined.

The given function is:

y = √x+4

It is a radical function. The function is defined if the expression inside the radical greater than or equal to zero.

x+4≥0

Subtract 4 from both the sides.

x+4-4 ≥ 0-4

x ≥ 4

The domain of the function is:

Domain = {xIx ≥ -4} = [-4,∞)

learn more about Domain here:

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3 years ago

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$4x^2 - 8x + 15 - \frac{78}{4x+5}$

Step-by-step explanation:

<em>To solve polynomial long division problems like these, it's helpful to build a long division table. Getting used to building these can make problems like this much simpler to solve.</em>

Begin by looking at the first term of the cubic polynomial.

What would we have to multiply $4x + 5$ by to get an expression containing $16x^3$ ? The answer is $4x^2$ , since $(4x + 5) \times 4x^2 = 16x^2 + 20x$ .

This is the first step of our long division, and we write out the start of our long division table like this:

${ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,4x^2\\4x + 5\quad)\!\!\overline{\,\,\,16x^3 - 12x^2 + 20x - 3}\\{ }\qquad{ }\quad{ }\quad{ }\,\,16x^3 + 20x^2\\$

On the left is the divisor. On top is $4x^2$ . In the middle is the polynomial we are dividing, and on the bottom is the result of multiplying our divisor by

The next step is to subtract the bottom expression from the middle one, like so:

${ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,4x^2\\4x + 5\quad)\!\!\overline{\,\,\,16x^3 - 12x^2 + 20x - 3}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\underline{16x^3 + 20x^2}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,0x^3 - 32x^2\\$

We are left with $-32x^2$ . The next thing to do is to add the next term of the polynomial we are dividing to the bottom line, like this:

Now we return to the beginning of the instructions, and repeat the process: namely, what would we have to multiply $4x + 5$ by to get an expression containing $-32x^2$ ? The answer is $-8x$ , and we fill out our long division table like so:

${ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,4x^2 - \,\,\,\,8x\\4x + 5\quad)\!\!\overline{\,\,\,16x^3 - 12x^2 + 20x - 3}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\underline{16x^3 + 20x^2}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 32x^2 + 20x\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 32x^2 - 40x\\$

Once again, we subtract the bottom expression from the one above it, and include the next term of the divisor, like so:

${ }\qquad{ }\qquad{ }\quad{ }4x^2 - \,\,\,\,8x \,+ 15\\4x + 5\quad)\!\!\overline{\,\,\,16x^3 - 12x^2 + 20x - 3}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\underline{16x^3 + 20x^2}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 32x^2 + 20x\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{- 32x^2 - 40x}\\{ }\qquad{ }\qquad{ }\qquad{ }\qquad{ }\qquad{ }\,\,\,\,\,60x - 3\\$

And repeat. What do we multiply $4x + 5$ by to get an expression containing $60x$ ? The answer is 15. Our completed long division table looks like this: ${ }\qquad{ }\qquad{ }\quad{ }4x^2 - \,\,\,\,8x \,+15\\4x + 5\quad)\!\!\overline{\,\,\,16x^3 - 12x^2 + 20x - 3}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\underline{16x^3 + 20x^2}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 32x^2 + 20x\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{- 32x^2 - 40x}\\{ }\qquad{ }\qquad{ }\qquad{ }\qquad{ }\qquad{ }\,\,\,\,\,60x - 3\\{ }\qquad{ }\hspace{3cm}\,\,\underline{60x + 75}\\{ }\hspace{4.3cm}\,\,-78$

Now, the expression at the top,

$4x^2 - 8x + 20x + 15$

is our quotient, and the last number, $-78$ , is our remainder.

Hence we arrive at the solution of

$\frac{16x^3-12x^2+20x-3}{4x+5} =4x^2 - 8x + 15 - \frac{78}{4x+5}.$

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