Let's solve both of these expressions step by step.

1. Finding $f(g(4))$:

• First, we need to find $g(4)$: $g(x) = -x + 6$ Substitute $x = 4$ into $g(x)$: $g(4) = -(4) + 6 = -4 + 6 = 2$
• Now, we use this result to find $f(g(4)) = f(2)$: $f(x) = 2x^2 - 3$ Substitute $x = 2$ into $f(x)$: $f(2) = 2(2)^2 - 3 = 2(4) - 3 = 8 - 3 = 5$ Thus, $f(g(4)) = 5$.

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2. Finding $g(f(4))$:

• First, we need to find $f(4)$: $f(x) = 2x^2 - 3$ Substitute $x = 4$ into $f(x)$: $f(4) = 2(4)^2 - 3 = 2(16) - 3 = 32 - 3 = 29$
• Now, we use this result to find $g(f(4)) = g(29)$: $g(x) = -x + 6$ Substitute $x = 29$ into $g(x)$: $g(29) = -(29) + 6 = -29 + 6 = -23$ Thus, $g(f(4)) = -23$.

Final Answers:

• $f(g(4)) = 5$
• $g(f(4)) = -23$

Do you want more details or have any questions?

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Here are 5 related questions:

1. What is the general form of $f(g(x))$ for these functions?
2. How would $f(g(x))$ change if $g(x) = -2x + 5$?
3. What is $f(f(4))$?
4. How would the results change if $f(x) = x^2 - 3x$?
5. Can we solve for $g(g(4))$?

Tip: Always solve functions step by step, ensuring substitution is done accurately for nested functions.