If

g(x) = x2 + 6x
with
x ≥ −3,
find
g−1(55)

I need help solving this problem this is an example but it does not thoroughly is explained. :/

you need x when g(x) = 55

x^2+6x=55
x^2+6x-55 = 0
(x+11)(x-5) = 0

Since the domain is x >= -3, that means that
g^-1(55) = 5

Note that specifying that x >= -3 means we have chosen only one branch of the parabola, whose vertex is at (-3,-9).

Since g(x) = x^2+6x,
x = -3 ±√(g+9)
we choose to use only the
x = -3+√(g+9)
Thus, when g=55, x=-3+√64 = -3+8 = 5

Ah, the infamous inverse function problem! Let's see if I can bring some humor to this equation.

To find the inverse of the function g(x) = x^2 + 6x, we need to swap the variables x and y and solve for y. So, let's start a magical swap dance!

g(x) = x^2 + 6x
Now, let's swap x and y:
x = y^2 + 6y.

Now we need to rearrange the equation to solve for y. We can do this by completing the square or using a magic wand. Let's go with the magic wand option because who doesn't love a touch of magic in math?

x = y^2 + 6y
(y^2 + 6y) + (9) = x + 9
(y^2 + 6y + 9) = x + 9
(y + 3)^2 = x + 9.

Now, let's use our comedic sense to simplify and solve for y.

Taking the square root of both sides, we find
y + 3 = ±√(x + 9).

Subtracting 3 from both sides, we have
y = -3 ± √(x + 9).

And there you have it! The inverse function of g(x). Now to find g^(-1)(55), we simply substitute 55 for x in the equation we just found.

g^(-1)(55) = -3 ± √(55 + 9).

So, g^(-1)(55) is approximately equal to -3 ± √64, which simplifies to -3 ± 8.

Hence, g^(-1)(55) can have two possible values: -3 + 8 = 5, or -3 - 8 = -11.

Remember, math can be funny too!

To find g−1(55), we need to find the inverse of g(x), which is denoted as g−1(x).

Step 1: Start by expressing g(x) in terms of y:
g(x) = x^2 + 6x = y

Step 2: Swap x and y in the equation:
x = y^2 + 6y

Step 3: Rearrange the equation to solve for y:
y^2 + 6y - x = 0

Step 4: This is now a quadratic equation. We can use the quadratic formula to find the solutions for y:
y = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = 1, b = 6, and c = -x.

Step 5: Plug in the value of x = 55 into the equation:
y = (-(6) ± √((6)^2 - 4(1)(-55))) / (2(1))

Simplifying further:
y = (-6 ± √(36 + 220)) / 2
y = (-6 ± √256) / 2
y = (-6 ± 16) / 2

Step 6: Solve for the two possible values of y:
y₁ = (-6 + 16) / 2 = 10 / 2 = 5
y₂ = (-6 - 16) / 2 = -22 / 2 = -11

Therefore, the two possible values for g−1(55) are 5 and -11.

To find the inverse of a function f(x), denoted as f^(-1)(x), you need to find a new function that "undoes" or reverses the original function.

In this case, we are given g(x) = x^2 + 6x and we need to find g^(-1)(55).
To find g^(-1)(55), we need to find the value of x such that g(x) = 55.

Step 1: Set up the equation:
g(x) = x^2 + 6x = 55.

Step 2: Rearrange the equation to isolate x:
x^2 + 6x - 55 = 0.

Step 3: Solve the equation by factoring or using the quadratic formula:
(x + 11)(x - 5) = 0.

The roots of this equation are x = -11 and x = 5.

Step 4: Determine which value of x satisfies the given condition x ≥ -3:
Since -11 is less than -3, it does not satisfy the condition. However, x = 5 is greater than or equal to -3, so it is a valid solution.

Therefore, g^(-1)(55) = 5.

In summary, to find the inverse of a function, set the function equal to the given value, solve for x, and then check if the solution satisfies any additional conditions or restrictions stated in the problem.