Use the functions f(x)=1/8x-3 and
g(x)=x^3 to find the indicated value or function.
(g^-1 X f^-1)(-3)
f^-1)(-3) is the value of x that you get when f(x) = -3. That value is 0.
g^-1(0) is the value of x that you get when g(x) = 0. That is also zero.
So (g^-1*f^-1)(-3) = 0
In functional form,
f^-1(x) = 8(x+3)
g^-1(x) = x^(1/3)
(g^-1*f^-1)(x) = [8(x+3)]^1/3
2(x+3)^1/3
To find the value of (g^-1 X f^-1)(-3), we need to first find the inverse functions of f(x) and g(x), and then substitute -3 into both inverse functions.
Let's start by finding the inverse function of f(x):
1. Start with f(x) = 1/8x - 3
2. Interchange x and y: x = 1/8y - 3
3. Solve for y: x + 3 = 1/8y
4. Multiply both sides by 8: 8(x + 3) = y
5. Simplify: y = 8x + 24
So, the inverse of f(x) is f^-1(x) = 8x + 24.
Now let's find the inverse function of g(x):
1. Start with g(x) = x^3
2. Interchange x and y: x = y^3
3. Take the cube root of both sides: y = ∛(x)
So, the inverse of g(x) is g^-1(x) = ∛(x).
Now we can substitute -3 into both inverse functions:
1. For f^-1(x), substitute x = -3: f^-1(-3) = 8(-3) + 24
2. Calculate: f^-1(-3) = -24 + 24 = 0
2. For g^-1(x), substitute x = -3: g^-1(-3) = ∛(-3)
3. Calculate: g^-1(-3) = -∛3
Finally, the value of (g^-1 X f^-1)(-3) is 0 times -∛3, which is 0.
So, (g^-1 X f^-1)(-3) = 0.
To find the value of (g^-1 X f^-1)(-3), we need to find the inverse functions of g(x) and f(x) first.
Let's start by finding the inverse of f(x):
Step 1: Substitute f(x) with y.
y = (1/8)x - 3
Step 2: Swap x and y.
x = (1/8)y - 3
Step 3: Solve for y.
x + 3 = (1/8)y
(8x + 24) = y
So, the inverse of f(x) is f^-1(x) = 8x + 24.
Now, let's find the inverse of g(x):
Step 1: Substitute g(x) with y.
y = x^3
Step 2: Swap x and y.
x = y^3
Step 3: Solve for y.
y = ∛x
So, the inverse of g(x) is g^-1(x) = ∛x.
Now, let's compute (g^-1 X f^-1)(-3) by plugging in -3 into g^-1(x) and then plugging that result into f^-1(x):
Step 1: Calculate g^-1(-3).
g^-1(-3) = ∛(-3) = -∛3
Step 2: Substitute -∛3 into f^-1(x).
f^-1(-∛3) = 8(-∛3) + 24
f^-1(-∛3) = -8∛3 + 24
Therefore, (g^-1 X f^-1)(-3) = -8∛3 + 24.