Let f(x)= 2x+1 and g(x) = 3x^2-4
find:
a) f^-1
b) gof (-2)
c) fog (x)
f^-1(x) = (x-1)/2
(g◦f)(-2) = g(f(-2)) = g(-3) = 23
(f◦g)(x) = f(g) = 2g+1 = 2(3x^2-4)+1 = 6x^2-7
To find the answers, let's work through the steps one by one:
a) To find the inverse of f(x), denoted as f^-1(x), we need to switch the roles of x and y in the equation f(x) = 2x + 1.
Step 1: Replace f(x) with y: y = 2x + 1.
Step 2: Swap x and y: x = 2y + 1.
Step 3: Solve for y. Subtract 1 from both sides of the equation to isolate 2y:
x - 1 = 2y.
Step 4: Divide both sides by 2 to solve for y:
y = (x - 1)/2.
Therefore, the inverse of f(x) is f^-1(x) = (x - 1)/2.
b) To find gof(-2), we substitute -2 into x in the equation g(x) = 3x^2 - 4.
g(x) = 3x^2 - 4
g(-2) = 3(-2)^2 - 4
g(-2) = 3(4) - 4
g(-2) = 12 - 4
g(-2) = 8.
Therefore, gof(-2) = 8.
c) To find fog(x), we substitute f(x) instead of x in the equation g(x) = 3x^2 - 4.
g(x) = 3x^2 - 4
g(f(x)) = 3(f(x))^2 - 4
g(f(x)) = 3(2x + 1)^2 - 4
g(f(x)) = 3(4x^2 + 4x + 1) - 4
g(f(x)) = 12x^2 + 12x + 3 - 4
g(f(x)) = 12x^2 + 12x - 1.
Therefore, fog(x) = 12x^2 + 12x - 1.
To find the answers, follow these steps:
a) To find the inverse function, f^-1, switch the x and y variables in the equation f(x) = 2x + 1, then solve for y.
f(x) = 2x + 1
y = 2x + 1
Now, switch the x and y variables:
x = 2y + 1
Solve for y:
x - 1 = 2y
y = (x - 1)/2
Therefore, the inverse function of f(x) = 2x + 1 is f^-1(x) = (x - 1)/2.
b) To find the value of gof(-2), substitute -2 into the function gof(x) = g(f(x)).
First, find the value of f(-2) by substituting -2 into f(x):
f(-2) = 2(-2) + 1 = -4 + 1 = -3
Now, substitute the value of f(-2) into g(f(x)):
g(f(-2)) = g(-3)
Substitute -3 into g(x):
g(-3) = 3(-3)^2 - 4 = 3(9) - 4 = 27 - 4 = 23
Therefore, gof(-2) = 23.
c) To find the value of fog(x), substitute x into the function fog(x) = f(g(x)).
First, find the value of g(x) by substituting x into g(x):
g(x) = 3x^2 - 4
Now, substitute the value of g(x) into f(g(x)):
f(g(x)) = f(3x^2 - 4)
Substitute 3x^2 - 4 into f(x):
f(3x^2 - 4) = 2(3x^2 - 4) + 1 = 6x^2 - 8 + 1 = 6x^2 - 7
Therefore, fog(x) = 6x^2 - 7.