Could you please check my answers and help me with two problems?
Determine which two functions are invereses of each other.
1. f(x)=5x, g(x)x/5, h(x)5/x
-I got that f(x) and g(x) are invereses of each other
2. f(x)=x-2/(2), g(x)=2x-2, h(x)=x+2/(2)
-I got that f(x) and h(x) are inverses of each other
3. f(x)x^4-3, g(x)= 4square root of x-3, h(x),=x^4+3
-I don't understand how to find the inverse of this problem.
To determine which two functions are inverses of each other, we can follow a simple process:
1. For a pair of functions to be inverses, the composition of one function with the other should give us the identity function.
2. The identity function, usually denoted as "I(x)" or "y=x," returns the same value that is input. In other words, I(x) = x.
Let's apply this process to the given problems:
1. f(x) = 5x, g(x) = x/5, h(x) = 5/x
To find the inverse of a function, we typically swap the "x" and "y," then solve for "y." Let's do that for each function:
- Inverse of f(x) = 5x:
Swap "x" and "y": x = 5y
Solve for "y": y = x/5 -> g(x) = x/5
- Inverse of g(x) = x/5:
Swap "x" and "y": x/5 = y
Solve for "y": y = (1/5)x -> f(x) = 5x
Therefore, f(x) and g(x) are indeed inverses of each other.
2. f(x) = (x-2)/2, g(x) = 2x-2, h(x) = (x+2)/2
Let's find inverses for each function:
- Inverse of f(x) = (x-2)/2:
Swap "x" and "y": x = (y-2)/2
Solve for "y": y = 2x + 2 -> g(x) = 2x + 2
- Inverse of h(x) = (x+2)/2:
Swap "x" and "y": x = (y+2)/2
Solve for "y": y = 2x - 2 -> h(x) = 2x - 2
Therefore, f(x) and h(x) are inverses of each other.
3. f(x) = x^4 - 3, g(x) = 4√(x-3), h(x) = x^4 + 3
To find the inverse of f(x), g(x), or h(x), we must follow a different approach. Unfortunately, none of these functions have an inverse that can be expressed using elementary functions (such as polynomials, exponentials, logarithms, etc.).
However, we can still determine if any of these pairs of functions are inverses by checking if their compositions yield the identity function. Given f(g(x)) or g(f(x)), if we can simplify to x, then they are inverses.
Let's calculate f(g(x)) and g(f(x)):
- f(g(x)) = f(4√(x-3)) = (4√(x-3))^4 - 3
- g(f(x)) = g(x^4 - 3) = 4√((x^4 - 3) - 3)
Simplifying these compositions doesn't yield the identity function, so neither f(x) and g(x) nor f(x) and h(x) are inverses.
I hope this helps clarify the concept of finding inverse functions! Let me know if you have any further questions.