$$ {\displaylines Låt\,(x)=3x+1,\,\,\,g(x)=ln(x2)\\ a)Bestäm\,maximaldefinitionsmänd\,för\,\,g(x).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,b)Bestäm\,\,g\,◦f(x)\,\,ochf^{-1}(x).\\ } $$

Let's go through the given problem step by step:

### a) Maximal Definitionsmängd for $ g(x) = \ln(x^2) $

The function $ g(x) = \ln(x^2) $ is a logarithmic function, and for a logarithmic function to be defined, its argument must be strictly greater than zero:

\[
x^2 > 0
\]

Since $ x^2 > 0 $ for all $ x \neq 0 $, the domain of $ g(x) $ is:

\[
x \in (-\infty, 0) \cup (0, \infty)
\]

In other words, the maximal domain (or definitionsmängd) of $ g(x) $ is all real numbers except $ x = 0 $.

**Maximal Definitionsmängd:**
\[
x \in \mathbb{R}, x \neq 0
\]

---

### b) $ g \circ f(x) $ and $ f^{-1}(x) $

#### 1. Compute $ g \circ f(x) $

The composition of functions $ g \circ f(x) $ means applying $ f(x) $ first, then applying $ g(x) $ to the result. We are given:

\[
f(x) = 3x + 1 \quad \text{and} \quad g(x) = \ln(x^2)
\]

Thus, $ g \circ f(x) = g(f(x)) $:

\[
g(f(x)) = \ln((f(x))^2) = \ln((3x + 1)^2)
\]

So:

\[
g \circ f(x) = \ln((3x + 1)^2)
\]

---

#### 2. Compute $ f^{-1}(x) $ (the inverse of $ f(x) $)

To find the inverse of $ f(x) = 3x + 1 $, we solve for $ x $ in terms of $ y $ where $ y = f(x) $:

\[
y = 3x + 1
\]

Solve for $ x $:

\[
y - 1 = 3x
\]

\[
x = \frac{y - 1}{3}
\]

Thus, the inverse function is:

\[
f^{-1}(x) = \frac{x - 1}{3}
\]

---

### Final Results:
a) The maximal definitionsmängd for $ g(x) $ is $ x \in \mathbb{R}, x \neq 0 $.

b) 
- $ g \circ f(x) = \ln((3x + 1)^2) $
- $ f^{-1}(x) = \frac{x - 1}{3} $

Would you like more details on any of these steps? Here are some related questions to deepen understanding:

1. What is the definition of the composition of functions?
2. How do you determine the domain of a composite function like $ g \circ f(x) $?
3. Why does $ x^2 $ remain positive for all $ x \neq 0 $?
4. How would you graph $ f(x) = 3x + 1 $ and its inverse $ f^{-1}(x) $?
5. What are the properties of logarithmic functions that help solve $ g(x) = \ln(x^2) $?

**Tip:** When finding the domain of a logarithmic function, always ensure the argument inside the logarithm is positive.

Låt(x)=3x+1, g(x)=ln(x^2)
a) Bestäm maximaldefinitionsmängd för g(x).
b) Bestäm g∘f(x) och f^{-1}(x).

Learn how to find the maximal domain of g(x) = ln(x^2), compose functions g∘f(x), and determine the inverse of f(x) = 3x + 1. This problem involves logarithmic properties, function composition, and the inverse function theorem, suitable for students in Grades 10-12.

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