y=5x^2+8
give the domain and range of the relation and indicate whether or not it is a function.
It's a function because every x-value that you plug in will have a single y-value associated with it.
DOMAIN (x-values): All Real Numbers
(since any x-value that you plug in will work)
RANGE (y-values): y>=8
(y will never be less than 8 because the y-intercept is 8 and the graph goes up from there)
so when you write is it going to be (8 or [8
range is y greater than or equal to 8
To find the domain and range of the relation, we need to understand what they represent in a function.
- The domain refers to the set of all possible input values (x-values) for the function. In other words, it represents the values for which the function is defined.
- The range refers to the set of all possible output values (y-values) for the function. In other words, it represents the values that the function can take on.
In this particular equation, y = 5x^2 + 8, we can see that it is a quadratic function. To determine the domain and range, we consider a few factors:
1. Domain:
Since there are no explicit restrictions or limitations on the input variable (x), the domain is considered to be all real numbers (-∞ to +∞). Therefore, the domain is (-∞, +∞).
2. Range:
For a quadratic function in the form y = ax^2 + bx + c, with a ≠ 0, the range can be determined by considering the vertex of the parabola. In this case, we have y = 5x^2 + 8, which is a U-shaped parabola opening upwards (since the coefficient of x^2 is positive).
Since the coefficient of x^2 is positive (a = 5 > 0), the parabola opens upward and the vertex represents the minimum point. The y-coordinate of the vertex (minimum point) can be calculated using the formula -b/(2a). Here, b = 0, so the x-coordinate of the vertex is 0 (since there is no x-term present) and the y-coordinate is 8. Hence, the vertex is (0, 8).
Since the vertex is the lowest point on the parabola, the range of the function is all real numbers greater than or equal to the y-coordinate of the vertex. Thus, the range is [8, +∞), where [8, ∞) means "from 8 to positive infinity, including 8".
3. Functionality:
To determine if the relation is a function, we need to check if each x-value (input) has a unique y-value (output). In this particular equation, we can see that for every x-value, there is only one corresponding y-value. Hence, the relation given by y = 5x^2 + 8 is a function.
To summarize:
- Domain: (-∞, +∞)
- Range: [8, +∞)
- Function: Yes, it is a function.