Could someone tell me how to find the inverse of g(x)= x^2-5x-6
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To find the inverse of the function g(x) = x^2 - 5x - 6, follow these steps:
Step 1: Start with the original equation: g(x) = x^2 - 5x - 6.
Step 2: Replace g(x) with y to represent the inverse function: y = x^2 - 5x - 6.
Step 3: Swap x and y to switch the roles of the variables: x = y^2 - 5y - 6.
Step 4: Rearrange the equation to solve for y. To do this, bring all terms to one side: y^2 - 5y - 6 - x = 0.
Step 5: Solve the quadratic equation to find the value(s) of y. You can use factoring, completing the square, or the quadratic formula. In this case, factoring is the most straightforward method.
The quadratic equation can be factored as (y - 6)(y + 1) = x + 1.
Step 6: Set each factor equal to zero and solve for y:
y - 6 = 0 or y + 1 = 0.
Solving these equations gives us y = 6 and y = -1.
Step 7: Now, we have two possible values for y. To find the inverse function, you need to swap the values of x and y:
When y = 6, x = 6.
When y = -1, x = -1.
Therefore, the inverse function of g(x) = x^2 - 5x - 6 is f(x) = 6 and f(x) = -1.
To find the inverse of a function, you need to follow a few steps:
1. Start with the given function, g(x).
2. Replace the function notation g(x) with y to get the equation in terms of y.
y = x^2 - 5x - 6
3. Swap x and y to interchange their positions in the equation.
x = y^2 - 5y - 6
4. Solve the equation from Step 3 for y, which means you need to rearrange the equation to isolate y.
y^2 - 5y - 6 - x = 0
This is a quadratic equation in terms of y.
5. Use factoring, completing the square, or the quadratic formula to solve for y. In this case, let's use the quadratic formula:
y = (-(-5) ± √((-5)^2 - 4(1)(-6 - x))) / (2(1))
Simplifying further,
y = (5 ± √(25 + 24x + 4x^2)) / 2
6. Simplify the equation and express the solutions separately.
y = (5 ± √(4x^2 + 24x + 25)) / 2
Now, the resulting equation is the inverse of g(x). You could express it as g^(-1)(x) = (5 ± √(4x^2 + 24x + 25)) / 2.
rewrite as
y = x^2 - 5x - 6
Step1: interchange the x and y variables
so
x = y^2 - 5y - 6
Step2: solve this new equation for y.
in this case you will have to solve it as a quadratic in y.
rewrite as
y^2 - 5y - 6 - x = 0
and a=1, b=-5 and c=(-6-x)
y = [5 ± √(25+4(6+x))]/2
= (5 ±√(49+4x))/2
g^-1(x) = (5 ±√(49+4x))/2
You can always test your answers by starting with any x, say x = 3
g(3) = 9 - 15 - 6 = -12
then putting that value of -12 back into the second relation should give you back the 3
g^-1(-12) = (5 ±√(49+4(-12)))/2
= (5 ± 1)/2 = 3 or 2
Notice that while the first was a function, the inverse is no longer a function.