Find 2 functions defined implicitly by the given relations:
(a) x² + y² = 25
(b) y² = x²
(a) y = (+or-) sqrt (25 - x^2)
(b) y = x or -x
Oh, I love a good math challenge! Let's find those implicitly defined functions.
(a) For the equation x² + y² = 25, we can rewrite it as y = ±sqrt(25 - x²). So, we have two functions: y = sqrt(25 - x²) and y = -sqrt(25 - x²). These functions represent the upper and lower halves of a circle with a radius of 5.
(b) Now, for the equation y² = x², we can rewrite it as y = ±x. So, we have two functions: y = x and y = -x. These functions represent a pair of straight lines that intersect at the origin and have a 45-degree angle.
So, there you have it! The two functions defined implicitly by the given relations are:
(a) y = sqrt(25 - x²) and y = -sqrt(25 - x²)
(b) y = x and y = -x
Now, go out there and have fun with those functions!
(a) To find two functions defined implicitly by the relation x² + y² = 25, we can solve for y in terms of x.
First, isolate y by subtracting x² from both sides:
y² = 25 - x²
Next, take the square root of both sides to solve for y:
y = ±√(25 - x²)
So, the two functions defined implicitly by the relation x² + y² = 25 are:
1. y = √(25 - x²)
2. y = -√(25 - x²)
(b) To find two functions defined implicitly by the relation y² = x², we can solve for y in terms of x.
By taking the square root of both sides, we get:
y = ±√(x²)
Simplifying, we have:
y = ±x
So, the two functions defined implicitly by the relation y² = x² are:
1. y = x
2. y = -x
To find two functions defined implicitly by the given relations, we can solve for one variable in terms of the other variable and then define a function using that equation. Let's solve each relation one by one:
(a) x² + y² = 25:
To solve for y in terms of x, we need to isolate y. We can subtract x² from both sides of the equation:
y² = 25 - x²
Now, we can take the square root of both sides to get rid of the square:
y = ± √(25 - x²)
Therefore, we have two functions defined implicitly by the relation x² + y² = 25:
1. f(x) = √(25 - x²)
2. g(x) = - √(25 - x²)
(b) y² = x²:
To solve for y in terms of x, we can take the square root of both sides:
y = ± √(x²)
Simplifying further, we get:
y = ± x
Therefore, we have two functions defined implicitly by the relation y² = x²:
1. f(x) = x
2. g(x) = -x
These are the two functions defined implicitly by the given relations.