I don't get how to do this question, please help!!
verify that r(r (x ) = r^-1(x ) if r( x) = 2x-7 / 4x-6
To verify that r(r(x)) = r^(-1)(x), we need to substitute the expression r(x) = (2x-7)/(4x-6) into both sides of the equation.
First, let's determine r(r(x)).
1. Substitute r(x) into r(r(x)): r(r(x)) = r((2x-7)/(4x-6)).
2. Simplify the expression inside the second r: r((2x-7)/(4x-6)) = r(2(2x-7)/2(2x-3)).
3. Cancel out the common factor in the numerator and denominator: r(2(2x-7)/2(2x-3)) = r((2x-7)/(2x-3)).
4. Substitute r(x) = (2x-7)/(4x-6) into the expression for r: r((2x-7)/(2x-3)) = [(2(2x-7)/(4x-6)]/(4(2x-7)/(4x-6)).
5. Simplify the expression inside r: [(2(2x-7)/(4x-6)]/(4(2x-7)/(4x-6)) = (2x-7)/(4x-6).
6. Simplify the expression further: (2x-7)/(4x-6) = r(x).
So, r(r(x)) simplifies to r(x).
Now, let's determine r^(-1)(x).
1. The inverse of r(x) is denoted as r^(-1)(x).
2. To find the inverse, switch the positions of x and r(x) and solve for x.
Starting with r(x) = (2x-7)/(4x-6),
Swap x and r(x): x = (2r(x)-7)/(4r(x)-6).
Solve for r(x) by multiplying both sides of the equation by (4r(x)-6):
x(4r(x)-6) = 2r(x)-7.
Distribute x: 4xr(x) - 6x = 2r(x) - 7.
Rearrange the equation: 4xr(x) - 2r(x) = 6x - 7.
Factor out r(x): r(x)(4x - 2) = 6x - 7.
Divide both sides by (4x - 2): r(x) = (6x - 7)/(4x - 2).
So, the inverse function r^(-1)(x) is (6x - 7)/(4x - 2).
Since r(r(x)) simplifies to r(x) and r^(-1)(x) is (6x - 7)/(4x - 2), we have verified that r(r(x)) = r^(-1)(x) for the given function r(x) = (2x-7)/(4x-6).