We are given two functions:

• $f(x) = 3x - 4$
• $g(x) = 2 - x^2$

We are asked to evaluate the following compositions:

(a) $(f \circ g)(x)$

The composition $(f \circ g)(x)$ means we apply $g(x)$ first and then apply $f$ to the result of $g(x)$. Mathematically:

$(f \circ g)(x) = f(g(x))$

Substitute $g(x) = 2 - x^2$ into $f(x)$:

$f(g(x)) = f(2 - x^2)$

Now, replace $x$ in $f(x) = 3x - 4$ with $2 - x^2$:

$f(2 - x^2) = 3(2 - x^2) - 4$ $= 6 - 3x^2 - 4$ $= 2 - 3x^2$

So, the result for $(f \circ g)(x)$ is:

$(f \circ g)(x) = 2 - 3x^2$

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(b) $(g \circ f)(x)$

The composition $(g \circ f)(x)$ means we apply $f(x)$ first and then apply $g$ to the result of $f(x)$. Mathematically:

$(g \circ f)(x) = g(f(x))$

Substitute $f(x) = 3x - 4$ into $g(x)$:

$g(f(x)) = g(3x - 4)$

Now, replace $x$ in $g(x) = 2 - x^2$ with $3x - 4$:

$g(3x - 4) = 2 - (3x - 4)^2$

First, expand $(3x - 4)^2$:

$(3x - 4)^2 = 9x^2 - 24x + 16$

Now, substitute this back into the equation for $g$:

$g(3x - 4) = 2 - (9x^2 - 24x + 16)$ $= 2 - 9x^2 + 24x - 16$ $= -9x^2 + 24x - 14$

So, the result for $(g \circ f)(x)$ is:

$(g \circ f)(x) = -9x^2 + 24x - 14$

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Final Answers:

• (a) $(f \circ g)(x) = 2 - 3x^2$
• (b) $(g \circ f)(x) = -9x^2 + 24x - 14$

Would you like more details or have any questions?

Related Questions:

1. How do you find the domain of a composition of functions?
2. What are the differences between function composition and function multiplication?
3. How do you handle compositions when one function involves a rational expression?
4. What happens if we compose functions that are undefined at certain points?
5. How do you determine the inverse of a composition of functions?

Tip:

When working with compositions of functions, always be careful with the order. $(f \circ g)(x)$ means applying $g$ first, then $f$, while $(g \circ f)(x)$ means applying $f$ first, then $g$.