f(x) = (x^2+3) and g(x) = x-1. find (fog)^-1

To find \((f \circ g)^{-1}(x)\), we need to follow these steps:

1. **Compose the functions \(f(x)\) and \(g(x)\):**
   \[
   (f \circ g)(x) = f(g(x)) = f(x - 1)
   \]
   Now, substitute \(g(x) = x - 1\) into \(f(x)\):
   \[
   f(g(x)) = (x - 1)^2 + 3
   \]
   So, \((f \circ g)(x) = (x - 1)^2 + 3\).

2. **Set \(y = (f \circ g)(x)\):**
   \[
   y = (x - 1)^2 + 3
   \]

3. **Solve for \(x\) in terms of \(y\):**
   \[
   y - 3 = (x - 1)^2
   \]
   \[
   \sqrt{y - 3} = |x - 1|
   \]
   This gives two possible solutions:
   \[
   x - 1 = \sqrt{y - 3} \quad \text{or} \quad x - 1 = -\sqrt{y - 3}
   \]
   Solving these:
   \[
   x = \sqrt{y - 3} + 1 \quad \text{or} \quad x = -\sqrt{y - 3} + 1
   \]

4. **Express \((f \circ g)^{-1}(x)\):**
   Since we have two possible expressions for \(x\), the inverse function \((f \circ g)^{-1}(x)\) is given by:
   \[
   (f \circ g)^{-1}(x) = 1 \pm \sqrt{x - 3}
   \]
   Here, \(1 + \sqrt{x - 3}\) typically represents the principal branch (positive root), but both expressions are valid depending on the domain.

### Do you want details or have any questions?
  
Here are 5 related questions:
1. How do you determine the domain of \((f \circ g)^{-1}(x)\)?
2. Can \((f \circ g)^{-1}(x)\) have a real output for any value of \(x\)?
3. What happens to \((f \circ g)^{-1}(x)\) when \(x = 3\)?
4. How do you compose two inverse functions, such as \((f^{-1} \circ g^{-1})(x)\)?
5. How does the graph of \((f \circ g)(x)\) compare to its inverse?

**Tip:** Always check the domain and range when working with inverse functions, as not all inputs may yield valid outputs.

f(x) = (x^2+3) and g(x) = x-1. Find (fog)^{-1}.

Discover how to find the inverse of the composed function (fog)^{-1} where f(x) = x^2 + 3 and g(x) = x - 1. This solution includes detailed steps for function composition, solving quadratic equations, and understanding inverse functions. Suitable for students in Grades 9-12.

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