Domain = the set of numbers that can be used as input (in this case, as values for x), and for which the function will return one (and only one) clear answer. For these values, the function is "well-defined"
In general, we assume all numbers are OK, then we look for problems, such as
-- is there a division? Then you must avoid dividing by zero (is there a value of x that causes a division by 0? such a value would be removed from the domain)
-- is there a square root? (or any even-numbered root) You cannot take the square root of a negative number. Any value of x that causes a square root of a negative number must be removed from the domain.
-- is there a logarithm in the function? You cannot take the log (in any base) of a negative number nor of zero.
However, in your function:
f(x) = -2x^2 + 10x + 2
Whatever number you use for x, you can always find the square of x, you can always multiply it by 2, and you can always add three numbers together.
Therefore, you will find that the domain of your function is "all real numbers" (from -infinity to +infinity) Be careful: the infinities themselves are NEVER part of the domain, since 'infinity' is not a well-defined number.
Range:
The set of all numbers that come up as output for the function, once you have tried all input values from the domain.
Here, because the leading coefficient (the -2 from -2x^2) is negative, the ends of the graph will go down to "minus infinity". Because there is only one curve in a second-degree graph (one less than the degree), there has to be a maximum.
Find that maximum value and the range cannot go higher than that.
Answers & Comments
f(x) = -2x² + 10x + 2
.....f(x) is CONCAVE DOWN because the coefficient
.....of x² is negative
.....The highest of f(x) will be at its vertex (2.5, 14.5)
SO,
Domain = (-∞,+∞)
Range = (-∞, 14.5]
(See graph below)
Domain = the set of numbers that can be used as input (in this case, as values for x), and for which the function will return one (and only one) clear answer. For these values, the function is "well-defined"
In general, we assume all numbers are OK, then we look for problems, such as
-- is there a division? Then you must avoid dividing by zero (is there a value of x that causes a division by 0? such a value would be removed from the domain)
-- is there a square root? (or any even-numbered root) You cannot take the square root of a negative number. Any value of x that causes a square root of a negative number must be removed from the domain.
-- is there a logarithm in the function? You cannot take the log (in any base) of a negative number nor of zero.
However, in your function:
f(x) = -2x^2 + 10x + 2
Whatever number you use for x, you can always find the square of x, you can always multiply it by 2, and you can always add three numbers together.
Therefore, you will find that the domain of your function is "all real numbers" (from -infinity to +infinity) Be careful: the infinities themselves are NEVER part of the domain, since 'infinity' is not a well-defined number.
Range:
The set of all numbers that come up as output for the function, once you have tried all input values from the domain.
Here, because the leading coefficient (the -2 from -2x^2) is negative, the ends of the graph will go down to "minus infinity". Because there is only one curve in a second-degree graph (one less than the degree), there has to be a maximum.
Find that maximum value and the range cannot go higher than that.
Domain 10, range (-1, 1)
domain: all real numbers
range: (-∞, -29/2]
vertex:
x = -10/(2 * -2)
x = 5/2
f(5/2) = 29/2
The parabola opens downward => y ≤ 29/2