$$ f(x)=2 x-1 $$ and $$ g(x)=3 x^{2} $$ Find both $$ f \circ g $$ and $$ g \circ f $$, stating domain and range of each.

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Given the functions:
$$ f(x) = 2x - 1 $$
$$ g(x) = 3x^{2} $$

Let's find both compositions $$ f \circ g $$ and $$ g \circ f $$, along with their domains and ranges.

### 1. Composition $$ f \circ g $$

**Definition:**
$$ (f \circ g)(x) = f(g(x)) $$

**Calculation:**
$$ f(g(x)) = f(3x^2) = 2(3x^2) - 1 = 6x^2 - 1 $$

**Domain:**
Both $$ f $$ and $$ g $$ are polynomial functions, which are defined for all real numbers. Therefore, the domain of $$ f \circ g $$ is:
$$ \text{Domain of } f \circ g: \mathbb{R} $$

**Range:**
$$ 6x^2 \geq 0 \quad \Rightarrow \quad 6x^2 - 1 \geq -1 $$
Thus, the range is:
$$ \text{Range of } f \circ g: [-1, \infty) $$

### 2. Composition $$ g \circ f $$

**Definition:**
$$ (g \circ f)(x) = g(f(x)) $$

**Calculation:**
$$ g(f(x)) = g(2x - 1) = 3(2x - 1)^2 $$
Expanding the square:
$$ 3(2x - 1)^2 = 3(4x^2 - 4x + 1) = 12x^2 - 12x + 3 $$

This can also be written in vertex form by completing the square:
$$ 12x^2 - 12x + 3 = 12\left(x^2 - x + \frac{1}{4}\right) - 3 + 3 = 12\left(x - \frac{1}{2}\right)^2 $$

**Domain:**
Again, since both $$ f $$ and $$ g $$ are defined for all real numbers, the domain of $$ g \circ f $$ is:
$$ \text{Domain of } g \circ f: \mathbb{R} $$

**Range:**
$$ \left(x - \frac{1}{2}\right)^2 \geq 0 \quad \Rightarrow \quad 12\left(x - \frac{1}{2}\right)^2 \geq 0 $$
Thus, the range is:
$$ \text{Range of } g \circ f: [0, \infty) $$

### Summary

- **$$ f \circ g $$(x):**
  $$ f \circ g (x) = 6x^2 - 1 $$
  $$ \text{Domain}: \mathbb{R} $$
  $$ \text{Range}: [-1, \infty) $$

- **$$ g \circ f $$(x):**
  $$ g \circ f (x) = 12x^2 - 12x + 3 $$
  $$ \text{Domain}: \mathbb{R} $$
  $$ \text{Range}: [0, \infty) $$

**Final Answer:**

After simplifying, the compositions are

f ∘ g (x ) = 6x² – 1 with domain all real numbers and range [ –1, ∞ )

g ∘ f (x ) = 12x² – 12x + 3 with domain all real numbers and range [ 0, ∞ )
$$ f \circ g(x) = 6x^2 - 1 $$ with domain all real numbers and range $$[-1, \infty)$$. $$ g \circ f(x) = 12x^2 - 12x + 3 $$ with domain all real numbers and range $$[0, \infty)$$.

and
Find both and , stating domain and
range of each.

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and Find both and , stating domain and range of each.