All inputs are valid. There is nothing you can put in for x that won't work.

Domain looks at x-values and range looks at y-values.

The x-values appear to continue to go on forever, which suggests the answer:

"all real numbers"

The y-values are all number that are equal to nine or less which is

So you answer is:

Domain: All real numbers

Range:

The range is the existing y-values that contains the function.

Notice that this is a parabola that opens downward, and the y-intercept is four.

This means that the highest y-value on this graph is four. The y-values will approach negative infinity as the domain, or x-values, approaches to positive and negative infinity.

The answer is:

Expand the quadratic.

Use the FOIL method to expand the binomials.

The equation becomes:

Now that we have the equation in format, find the vertex. This will determine the minimum of the parabola.

The formula is:

Substitute the values.

To find the y-value, substitute the x-value back to the original equation.

The minimum is:

Because the value of is positive, the parabola will open up.

The range is:

The expression under the square root symbol cannot be negative, so to find the domain, set that expression .

The domain includes all x-values less than or equal to 7, which can be written as .

A function has to pass the vertical line test, which means that a vertical line can only cross the function one time. To put it another way, for any given value of , there can only be one value of . For the function , there is one value for two possible values. For instance, if , then . But if , as well. This function fails the vertical line test. The other functions listed are a line,, the top half of a right facing parabola, , a cubic equation, , and a semicircle, . These will all pass the vertical line test.

A basic knowledge of absolute value and its functions is valuable for this problem. However, if you do not know what the typical shape of an absoluate value function looks like, one can always plug in values and plot points.

Upon doing so, we learn that the -values (domain) are not restricted on either end of the function, creating a domain of negative infinity to postive infinity.

If we plug in -100000 for , we get 100000 for .

If we plug in 100000 for , we get 100000 for .

Additionally, if we plug in any value for , we will see that we always get a real, defined value for .

**Extra Note: Due to the absolute value notation, the negative (-) next to the is not important, in that it will always be made positive by the absolute value, making this function the same as . If the negative (-) was outside of the absolute value, this would flip the function, making all corresponding -values negative. However, this knowledge is most important for range, rather than domain.

Domain is finding the acceptable values that will make the function generate real values. is a quadratic function therefore any values will always generate real values. Answer is all real numbers.

The domain is the set of all potential x-values. In this case, the graph starts at and never extends to the left of this point. This means that , inclusive. The range is the set of all potential y-values. In this case, there are no restrictions on the y-axis, so the range is the set of all real numbers.

The domain is the set of possible value for the variable. We can find the impossible values of by setting the denominator of the fractional function equal to zero, as this would yield an impossible equation.

Now we can solve for .

There is no real value of that will fit this equation; any real value squared will be a positive number.

The radicand is always positive, and is defined for all real values of . This makes the domain of the set of all real numbers.