Layton made 117$ for 9 hours of work.at the same rate,how many hours would she have to work to make $65?
1 answer:
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Answer:
The equation determine a relation between x and y
x = ±
y = ±
The domain is 1 ≤ y ≤ 3
The domain is -1 ≤ x ≤ 1
The graphs of these two function are half circle with center (0 , 2)
All of the points on the circle that have distance 1 from point (0 , 2)
Step-by-step explanation:
* Lets explain how to solve the problem
- The equation x² + (y - 2)² and the relation "(x , y) R (0, 2)", where
R is read as "has distance 1 of"
- This relation can also be read as “the point (x, y) is on the circle
of radius 1 with center (0, 2)”
- “(x, y) satisfies this equation , if and only if, (x, y) R (0, 2)”
* <em>Lets solve the problem</em>
- The equation of a circle of center (h , k) and radius r is
(x - h)² + (y - k)² = r²
∵ The center of the circle is (0 , 2)
∴ h = 0 and k = 2
∵ The radius is 1
∴ r = 1
∴ The equation is ⇒ (x - 0)² + (y - 2)² = 1²
∴ The equation is ⇒ x² + (y - 2)² = 1
∵ A circle represents the graph of a relation
∴ The equation determine a relation between x and y
* Lets prove that x=g(y)
- To do that find x in terms of y by separate x in side and all other
in the other side
∵ x² + (y - 2)² = 1
- Subtract (y - 2)² from both sides
∴ x² = 1 - (y - 2)²
- Take square root for both sides
∴ x = ±
∴ x = g(y)
* Lets prove that y=h(x)
- To do that find y in terms of x by separate y in side and all other
in the other side
∵ x² + (y - 2)² = 1
- Subtract x² from both sides
∴ (y - 2)² = 1 - x²
- Take square root for both sides
∴ y - 2 = ±
- Add 2 for both sides
∴ y = ±
∴ y = h(x)
- In the function x = ±
∵ ≥ 0
∴ 1 - (y - 2)² ≥ 0
- Add (y - 2)² to both sides
∴ 1 ≥ (y - 2)²
- Take √ for both sides
∴ 1 ≥ y - 2 ≥ -1
- Add 2 for both sides
∴ 3 ≥ y ≥ 1
∴ The domain is 1 ≤ y ≤ 3
- In the function y = ±
∵ ≥ 0
∴ 1 - x² ≥ 0
- Add x² for both sides
∴ 1 ≥ x²
- Take √ for both sides
∴ 1 ≥ x ≥ -1
∴ The domain is -1 ≤ x ≤ 1
* The graphs of these two function are half circle with center (0 , 2)
* All of the points on the circle that have distance 1 from point (0 , 2)
Composing functions means that the input of the outer functions is the output of the inner function.
In fact, you can rewrite the circle notation as
So, we can substitute g(x) with its expression:
And since f(x)=x+5, we simply have to add 5 to its input:
Similarly, we have, substituting f with its expression,
And since g(x)=4x+2, we have to multiply the input by 4 and add 2: