Use the functions f(x)=x+4 and g(x)=2x-5 to find the specified functions.
(g x f)^-1
Please state in words your notation
(g x f)^-1 I assume you do not mean the reciprocal of g times f.
let y = g x f
= (x+4)(2x-5)
y = 2x^2 + 3x - 20
so the inverse of that is
x = 2y^2 + 3y - 20
2y^2 + 3y - 20 - x = 0
which by the way is not a function
to solve for y you will have to use the quadratic formula and
y = (-3 ± √(9-8(-20-x))/4
etc.
To find the specified function (g x f)^-1, we need to first find the composition of the two functions g and f, and then take the inverse of the resulting function.
The composition of two functions g and f is denoted as (g o f)(x), and it means that we first apply f to x and then apply g to the result.
So, let's find (g o f)(x):
(g o f)(x) = g(f(x))
First, apply f to x:
f(x) = x + 4
Now, apply g to the result:
g(x + 4) = 2(x + 4) - 5
= 2x + 8 - 5
= 2x + 3
So, (g o f)(x) = 2x + 3
Now, to find the inverse of (g o f)(x), denoted as ((g o f)(x))^-1, we need to switch the roles of x and y and solve for y.
So, let's switch x and y:
x = 2y + 3
Next, solve for y:
x - 3 = 2y
(your school's answer probably here) (x - 3) / 2 = y
Therefore, the specified function ((g o f)(x))^-1 is:
((g o f)(x))^-1 = (x - 3) / 2