If
g(x) = x2 + 8x
with
x ≥ −4,
find
g−1(20)
when does g(x)=20?
When x=2
So, g^-1(20) = 2
Note that g(-10) = 20 as well, but that does not matter, since we have restricted our domain to x >= -4.
Well, well, well, aren't we feeling a bit mathematical today? Alright, let's see what we can do here.
To find g^(-1)(20), we need to find the value of x that makes g(x) equal to 20. So, let's set up the equation:
x^2 + 8x = 20
Now, we can solve this equation. But before we dive into crunching numbers, let me ask you, have you ever heard the joke about the mathematician who got lost in a forest? He used a Taylor series to approximate his way out. Clever, huh?
Anyway, let's solve the equation. Rearrange it and set it to zero:
x^2 + 8x - 20 = 0
Now, we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values from our equation, we get:
x = (-(8) ± √((8)^2 - 4(1)(-20))) / (2(1))
And after some numerical hocus-pocus, we find:
x = -10 or x = 2
So, for g^(-1)(20), we have two possible solutions: x = -10 or x = 2. Time to put a smile on that face and celebrate the mathiness! 🤡
To find g^-1(20), we need to find the inverse function of g(x) = x^2 + 8x first.
Step 1: Replace g(x) with y.
y = x^2 + 8x
Step 2: Swap x and y.
x = y^2 + 8y
Step 3: Solve the quadratic equation for y using the quadratic formula.
y^2 + 8y - x = 0
The quadratic formula is:
y = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 8, c = -x. Substituting these values into the quadratic formula, we get:
y = (-8 ± √(8^2 - 4(1)(-x))) / (2(1))
Simplifying it further:
y = (-8 ± √(64 + 4x)) / 2
y = (-8 ± √(4(x + 16))) / 2
y = (-8 ± 2√(x + 16)) / 2
y = -4 ± √(x + 16)
Step 4: Replace y with g^-1(x).
g^-1(x) = -4 ± √(x + 16)
Now, we can find g^-1(20).
g^-1(20) = -4 ± √(20 + 16)
g^-1(20) = -4 ± √36
g^-1(20) = -4 ± 6
This gives us two potential values for g^-1(20):
1. g^-1(20) = -4 + 6 = 2
2. g^-1(20) = -4 - 6 = -10
Therefore, g^-1(20) can be either 2 or -10.
To find g^(-1)(20), we need to find the inverse function of g(x), which means we need to solve for x in terms of y.
First, let's rewrite g(x):
g(x) = x^2 + 8x
Now, replace g(x) with y:
y = x^2 + 8x
To find the inverse, we need to interchange the x and y variables:
x = y^2 + 8y
Next, rearrange the equation to solve for y:
0 = y^2 + 8y - x
Now, we have a quadratic equation in terms of y. To solve for y, we can use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 8, and c = -x. Substituting these values into the quadratic formula, we get:
y = (-8 ± √(8^2 - 4(1)(-x))) / 2(1)
y = (-8 ± √(64 + 4x)) / 2
y = (-8 ± √(4x + 64)) / 2
Since we're given x ≥ -4, we can disregard the negative solution because it would result in a negative value for x. Therefore, we only consider the positive solution:
y = (-8 + √(4x + 64)) / 2
y = (√(4x + 64) - 8) / 2
y = (√(4x + 64) - 8) / 2
Now, we have expressed the inverse function of g(x) in terms of x.
To find g^(-1)(20), substitute y with 20 in the equation:
20 = (√(4x + 64) - 8) / 2
Now, solve for x.
Multiply both sides of the equation by 2:
40 = √(4x + 64) - 8
Add 8 to both sides of the equation:
48 = √(4x + 64)
Square both sides of the equation:
2304 = 4x + 64
Subtract 64 from both sides of the equation:
2240 = 4x
Divide both sides of the equation by 4:
x = 560
Therefore, g^(-1)(20) = 560.