Prove if the given functions are inverses by evaluating the composition f(h(x)) or h(f(x))
g(x) = -(4/-x-3) +1
f(x)= (3/-x-1) -2
How do we go about once we for example plug in the first expression into x in the second expression, in terms of what to do exactly.
Thanks
First, make sure you have typed things correctly. What you have written is gibberish. So, I shall go with
g(x) = -4/(-x-3) + 1
First of all, what's with all those - signs? Oh, well ...
swap variables and solve for x:
h(x) = 4/(x-1) - 3 = (7-3x)/(x-1)
By definition, f^-1(f(x)) = f(f^-1(x)) = x
So, to show that g and h are inverses, just evaluate g(h) and h(g)
g(h(x)) = -4/(-h(x)-3) + 1
= -4/(-(7-3x)/(x-1)-3)+1
= 4/((7-3x)/(x-1)+3)+1
= 4/((7-3x+3x-3))/(x-1))+1
= 4/(4/(x-1))+1
= 4 * (x-1)/4 + 1
= x-1+1
= x
Work h(g(x)) the same way
After you fix up your functions with the proper use of parentheses, of course!
oh. sorry, what exactly happened at the “swap variables and solve for x” step? How did you get (7-3x)/(x-1)
Ooops my bad. If you swap first, solve for y
you need to review how to find an inverse function.
If y = 2x-3
to find f^-1, first swap variables
x = 2y-3
then solve for y
y = (x+3)/2
Or, you could solve for x and then swap
y = 2x-3
y-3 = 2x
(y-3)/2 = x
x = y^-1 = (x-3)/2
y+3
To determine if two functions are inverses, we need to evaluate the composition of the functions. Evaluating the composition involves plugging one function into the other and simplifying the expression.
Let's start by evaluating f(h(x)):
1. Replace x in f(x) with h(x):
f(h(x)) = (3/(-(4/(-x-3)) + 1) - 2
To simplify this expression, we need to follow the order of operations (parentheses, exponents, multiplication/division, and addition/subtraction). Let's start by simplifying the innermost expression, -(4/(-x-3)):
2. Simplify -(4/(-x-3)):
Multiplying a negative and positive sign gives us a negative result, so -(4/(-x-3)) becomes (4/(-x-3)).
Now, we can substitute this simplified expression back into f(h(x)):
f(h(x)) = (3/(4/(-x-3)) + 1) - 2
3. Simplify the division:
Dividing by a fraction is the same as multiplying by its reciprocal. So, (3/(4/(-x-3))) becomes (3 * (-x-3)/4).
Now, substituting this back into f(h(x)):
f(h(x)) = (3 * (-x-3)/4 + 1) - 2
4. Simplify the expression:
Now, distribute the 3/4 to (-x-3):
f(h(x)) = (-3x - 9)/4 + 1 - 2
Combining like terms:
f(h(x)) = (-3x - 9 + 4 - 8)/4
Simplifying further:
f(h(x)) = (-3x - 13)/4
Now, let's evaluate h(f(x)):
1. Replace x in h(x) with f(x):
h(f(x)) = -(4/(-(3/(-x-1)) - 2)) + 1
Similarly, follow the order of operations to simplify this expression:
2. Simplify -(3/(-x-1)):
Multiplying a negative and a positive sign gives us a negative result, so -(3/(-x-1)) becomes (3/(-x-1)).
Now, substitute this expression back into h(f(x)):
h(f(x)) = -(4/(3/(-x-1)) - 2) + 1
3. Simplify the division:
Again, dividing by a fraction is the same as multiplying by its reciprocal. (4/(3/(-x-1))) becomes (4 * (-x-1)/3).
Substituting this back into h(f(x)):
h(f(x)) = -(4 * (-x-1)/3 - 2) + 1
4. Simplify the expression:
Distribute the -4/3 to (-x-1):
h(f(x)) = -((-4/3) * (-x - 1) - 2) + 1
Simplifying further:
h(f(x)) = -((-4/3)(-x) - (-4/3)(1) - 2) + 1
Simplifying further:
h(f(x)) = -((4/3)x + 4/3 - 2) + 1
Combining like terms:
h(f(x)) = -(4/3)x - 2/3 + 1
Simplifying:
h(f(x)) = -(4/3)x - 2/3 + 3/3
Combining like terms again:
h(f(x)) = -(4/3)x + 1/3
After evaluating the compositions f(h(x)) and h(f(x)), we can see that f(h(x)) = (-3x - 13)/4 and h(f(x)) = -(4/3)x + 1/3. Since these are not equal, the given functions g(x) and f(x) are not inverses of each other.