Find the domain and range for y=x^2+9.
Is the answer all real numbers? How would you enter that in interval notation? From Googling the definition of real numbers, I now understand that the infinity sign would not be included. For a continuous function you use closed brackets, but I just don't know what numbers to put into the brackets.
Is factoring something I should do here? I'm not sure.
for any x, x^2 >= 0
so, y >= 9 for all real x.
x can, of course, be any real number.
D: (-∞,+∞)
R: [9,+∞)
Thank you so much! That really helps. I totally forgot that the range would have to be over 9 due to the plus 9 in the equation.
To find the domain and range of the function y = x^2 + 9, we will consider the real numbers as the input (x) and the corresponding output (y).
Domain:
The domain of a function represents all the possible values that the input (x) can take. For a polynomial function like y = x^2 + 9, there are no restrictions on the values that x can take. Therefore, the domain of this function is all real numbers (-∞, ∞).
Range:
To find the range of a function, we need to determine all the possible values that the output (y) can take. For the given function, y = x^2 + 9, let's consider a few points:
When x = 0, y = 0^2 + 9 = 9.
When x = 1, y = 1^2 + 9 = 10.
When x = -1, y = (-1)^2 + 9 = 10.
From these calculations, we can see that the function y = x^2 + 9 always yields a positive value for y. Since there are no restrictions on the values y can take, the range of this function is all real numbers greater than or equal to 9, or, in interval notation, [9, ∞).
Factoring is not necessary for finding the domain and range of this particular function since there are no factors that restrict its values.
To find the domain and range for the equation y = x^2 + 9, let's start with the domain.
The domain refers to the set of possible values for the independent variable (in this case, x) for which the function is defined. Since the equation is a polynomial and does not involve any square roots or fractions with denominators, there are no specific restrictions on the domain. Therefore, the domain of y = x^2 + 9 is all real numbers.
In interval notation, to represent all real numbers, we can use the interval (-∞, +∞). The negative infinity symbol, -∞, indicates that the values can extend indefinitely in the negative direction, and the positive infinity symbol, +∞, indicates that the values can extend indefinitely in the positive direction. The parentheses () are used to indicate that infinity itself is not included in the interval.
Now let's determine the range.
The range refers to the set of possible values for the dependent variable (in this case, y) that the function can take on. Since the equation y = x^2 + 9 is a quadratic function with a positive coefficient for x^2, it means the parabola opens upwards. This means that the minimum value of y will be at its vertex. In this case, the vertex of the parabola is at the point (0, 9).
Therefore, the minimum value of y is 9, and there is no maximum value. So the range of the equation y = x^2 + 9 is all real numbers greater than or equal to 9.
In interval notation, we can represent the range as [9, +∞), using a square bracket [ for 9 to indicate that it is included in the interval, and a parenthesis ) for +∞ to indicate that infinity itself is not included.
Regarding factoring, it is not required to find the domain and range of this particular equation. Factoring is typically used to find the x-intercepts (zeros) of a quadratic equation. However, in this case, we are specifically focused on the domain (possible x values) and range (possible y values), not on the roots of the equation.