Respuesta :
Answer:
1) The relation is a function.
2) The relation is a function.
- Domain: -∞ < x < ∞ , or (-∞, ∞)
- Range: (-∞, ∞), or -∞ < f(x) < ∞, or (-∞, ∞)
3) y does not vary with x (no constant of variation).
4) [tex]\LARGE\mathsf{Slope\:(m) = -\frac {4}{9}}[/tex]
7) y = f(x - 2) - 7
Step-by-step explanation:
Note:
I will be providing solutions for questions 1, 2, 3, 4, and 7, in accordance with Brainly's rules.
Question 1:
Determine whether each relation is a function. Find the domain and range. {(10,2), ( -10,2), (6,4), (5,3, (-6,7)}
A given relation is a function such that no two ordered pairs have the same input and have different outputs.
In the given relation, each input value corresponds exactly to one output value. Therefore, the given relation is a function.
Question 2:
Determine whether each relation is a function. Find the domain and range.
f(x) = -x +4
The given linear function, f(x) = -x +4, is a function, as each x-coordinates along the line has its own corresponding y-coordinates. Therefore, the given relation is a function.
The domain of the linear function is all real numbers, as there are no limiting factors or constraints that will make the function undefined.
Similarly, the range of the given linear function is all real numbers.
Domain: (-∞, ∞), or -∞ < x < ∞
Range: (-∞, ∞), or -∞ < f(x) < ∞
Question 3:
For the function below, determine whether y varies with x. If so, find the constant of variation and write the function rule. {(-2,3), (1,4), (2,7)}
To determine the constant of variation of the given relation, we could use the slope formula:
(x₁, y₁) = (-2,3)
(x₂, y₂) = (1,4)
[tex]\LARGE\mathsf{m = \frac {(y_2\: -\: y_1)}{(x_2\: -\: x_1)}}\\\LARGE\mathsf{m = \frac {4\: -\: (-3)}{1\: -\: (-2)}\:=\frac{4\:+\:3}{1\:+\:2}\:=\:\frac{7}{3}}[/tex]
Next:
(x₁, y₁) = (1,4)
(x₂, y₂) = (2,7)
[tex]\LARGE\mathsf{m = \frac {(y_2\: -\: y_1)}{(x_2\: -\: x_1)}}\\\LARGE\mathsf{m = \frac {7\: -\:4}{2\: -\:1}\:=\frac{3}{1}\:=\:3}[/tex]
Since our solutions resulted in varying slopes, then it means that the given points do not have a constant of variation. Thus, we cannot write a function rule.
Question 4:
Identify the slope of the line that passes through the given point. (5, 2) and (-4, 6).
Let (x₁, y₁) = (5, 2)
(x₂, y₂) = (-4, 6)
Substitute these values into the slope formula:
[tex]\LARGE\mathsf{m = \frac {(y_2\: -\: y_1)}{(x_2\: -\: x_1)}}\\\LARGE\mathsf{m = \frac {6\: -\:2}{-4\: -\:5}\:=\frac{4}{-9}\:=\:-\frac{4}{9}}[/tex]
Thus, the slope of the line is: [tex]\LARGE\mathsf{m = -\frac {4}{9}}[/tex].
Question 7:
Write the equation for the transformation of the graph of y = f(x)
translated 2 units left and 7 units down.
Given the parent function, y = f(x), where it is translated 2 units to the left, and is shifted 7 units down.
The horizontal translation of the parent graph |h | units to the left is represented by y = f(x - h), where h > 0. Since h = 2, then simply substitute the value of h into the transformed function: y = f(x - 2).
Next, the vertical translation of the parent graph by k units downward is represented by y = f(x) - k, where k < 0. Since k = 7, then simply substitute the value of k into the transformed function: y = f(x - 2) - 7.
Therefore, the equation of the transformed graph is: y = f(x - 2) - 7.
