We are given the following functions:

• $f(x) = \sqrt{x^2 - 2x + 6}$
• $g(x) = 7x + 3$

1. Finding $(f \circ g)(x)$:

The composition $(f \circ g)(x)$ means we need to substitute $g(x)$ into $f(x)$, so:

$(f \circ g)(x) = f(g(x)) = f(7x + 3)$

Substitute $7x + 3$ into the expression for $f(x)$:

$f(7x + 3) = \sqrt{(7x + 3)^2 - 2(7x + 3) + 6}$

Now, simplify step by step:

• First, expand $(7x + 3)^2$: $(7x + 3)^2 = 49x^2 + 42x + 9$
• Next, expand $-2(7x + 3)$: $-2(7x + 3) = -14x - 6$
• Now, substitute these back into the square root: $f(7x + 3) = \sqrt{49x^2 + 42x + 9 - 14x - 6 + 6}$ Simplifying the terms: $f(7x + 3) = \sqrt{49x^2 + 28x + 9}$

2. Finding $(f \circ g)(-3)$:

Now, substitute $x = -3$ into $(f \circ g)(x)$:

$(f \circ g)(-3) = \sqrt{49(-3)^2 + 28(-3) + 9}$ First, calculate each term:

$49(-3)^2 = 49(9) = 441$ $28(-3) = -84$ $441 - 84 + 9 = 366$

So:

$(f \circ g)(-3) = \sqrt{366}$

Thus, $(f \circ g)(-3) \approx 19.13$ (since $\sqrt{366} \approx 19.13$).

Would you like any additional details or clarification?

Here are 5 related questions for further exploration:

1. How do we verify the domain of the composite function $f \circ g$?
2. What are the steps for calculating the derivative of $f(x) = \sqrt{x^2 - 2x + 6}$?
3. How would you evaluate $(f \circ g)(x)$ for a different value, say $x = 2$?
4. How do the graphs of $f(x)$, $g(x)$, and $f \circ g(x)$ compare?
5. What if $g(x)$ was a quadratic function instead of a linear one?

Tip: When performing function composition, ensure the input into the outer function is valid for the function's domain!