For each of the functions below, state the domain and range, the restrictions, the intervals of increasing and decreasing, the roots, y-intercepts, and vertices.
a)
f(x)=2x(^2)−8
b)
f(x)=+√x-2
domain of all exponentials is (-∞,∞)
Range of a^x is (0,∞), so I'm sure you can adjust that as needed here.
you know that the domain of √x is [0,∞), so now you just have to adjust to make sure that x-2 >= 0.
As for the range, just recall that square roots are never negative.
To determine the domain and range, restrictions, intervals of increasing and decreasing, roots, y-intercepts, and vertices for each function, let's analyze them one by one:
a) f(x) = 2x^2 - 8
Domain and Range:
- The domain is the set of all possible input values that x can take. In this case, since there are no restrictions on x, the domain is (-∞, +∞), which means all real numbers.
- The range is the set of all possible output values that f(x) can take. Since the function is quadratic and the coefficient of x^2 is positive (2), the parabola opens upwards, and the range is [c, +∞), where c is the y-coordinate of the vertex.
Restrictions:
- There are no restrictions or conditions mentioned in this function.
Intervals of Increasing and Decreasing:
- Since the coefficient of x^2 is positive, the parabola opens upwards, indicating that the function is increasing for all real values of x.
Roots:
- The roots, also known as the x-intercepts or zeros, are the values of x for which f(x) = 0. To find the roots, we set f(x) = 2x^2 - 8 equal to 0 and solve for x. In this case, we have:
2x^2 - 8 = 0
x^2 - 4 = 0
(x - 2)(x + 2) = 0
x = 2 or x = -2
So, the roots are x = 2 and x = -2.
Y-intercept:
- The y-intercept is the point where the graph intersects the y-axis. To find the y-intercept, we set x = 0 in the function f(x). In this case, we have:
f(0) = 2(0)^2 - 8
f(0) = -8
So, the y-intercept is (0, -8).
Vertex:
- The vertex represents the highest or lowest point on the parabola. To find the vertex, we use the formula x = -b / (2a), where a and b are the coefficients of the quadratic function. In this case, a = 2 and b = 0, so the x-coordinate of the vertex is x = -0 / (2 * 2) = 0. Then we substitute this value of x back into the function to find the y-coordinate:
f(0) = 2(0)^2 - 8
f(0) = -8
So, the vertex is (0, -8).
b) f(x) = √(x - 2)
Domain and Range:
- The domain represents the set of all possible input values that x can take. However, in the case of a square root function (√), the value inside the square root must be greater than or equal to zero to avoid taking the square root of a negative number. Therefore, x - 2 ≥ 0, which means that x ≥ 2. Hence, the domain is [2, +∞).
- The range represents the set of all possible output values that f(x) can take. Since the function involves the square root (√), the range is [0, +∞), meaning it can only output non-negative real numbers or zero.
Restrictions:
- The restriction is that the value inside the square root (√) must be greater than or equal to zero, as mentioned before. Therefore, x - 2 ≥ 0, or x ≥ 2.
Intervals of Increasing and Decreasing:
- Since the function involves a square root (√), it is always increasing. However, it is important to note that the function is only defined for x ≥ 2.
Roots:
- To find the roots or x-intercepts, we set f(x) = √(x - 2) equal to 0. However, a square root function can never equal zero (since √x > 0 for all x), so there are no roots in this case.
Y-intercept:
- The y-intercept is the point where the graph intersects the y-axis. To find it, we set x = 0 in the function f(x). In this case, we have:
f(0) = √(0 - 2)
f(0) = √(-2)
The value inside the square root (√) is negative, so the y-intercept is undefined.
Vertex:
- The square root function (√) does not have a vertex like a quadratic function. The vertex concept is specific to parabolic shapes and is not applicable to square root functions.
Note: It's always important to consider the specific characteristics and properties of each function to determine its domain, range, restrictions, intervals of increasing and decreasing, roots, y-intercepts, and vertices.