Two functions g and h are defined as :
g:x--> 2x+3/x-4
h:x --> 1/x
Write expressions for :
a) hg(x)
b) g inverse 1(x)
To write the expressions for hg(x) and g inverse 1(x), we need to perform the composition of the two functions g and h, as well as find the inverse of function g.
a) To find hg(x), we need to substitute the expression for h into the function g.
First, let's find the expression for h(g(x)), where g(x) is substituted into h(x):
h(g(x)) = h(2x + 3)/(x - 4)
Now, let's substitute the expression for h(x):
h(g(x)) = 1 / (2x + 3)/(x - 4)
We can simplify this further by multiplying the numerator and denominator by the reciprocal of the fraction in the denominator:
h(g(x)) = 1 / (2x + 3) * (x - 4)
b) To find the expression for the inverse of g, denoted as g inverse 1(x), we need to interchange the x and y variables and solve for y.
Start by writing the equation for the function g(x):
g(x) = (2x + 3) / (x - 4)
Now, interchange x and y:
x = (2y + 3) / (y - 4)
To solve for y, cross-multiply:
x(y - 4) = 2y + 3
xy - 4x = 2y + 3
xy - 2y = 4x + 3
Factor out y on the left side:
y(x - 2) = 4x + 3
Finally, divide both sides by (x - 2):
y = (4x + 3) / (x - 2)
Therefore, the expression for g inverse 1(x) is:
g inverse 1(x) = (4x + 3) / (x - 2)