Given the functions $f(x) = x^2$ and $g(x) = x - 5$, let's solve the problems step by step.

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

The notation $(f \circ g)(x)$ represents the composition of $f$ and $g$, meaning $f(g(x))$. This means we will substitute $g(x)$ into the function $f(x)$.

1. $g(x) = x - 5$
2. Now substitute $g(x)$ into $f(x)$: $f(g(x)) = f(x - 5) = (x - 5)^2$
3. Simplify the expression: $(x - 5)^2 = x^2 - 10x + 25$ So, $(f \circ g)(x) = x^2 - 10x + 25$.

Part b: $(g \circ f)(3)$

The notation $(g \circ f)(3)$ represents the composition $g(f(3))$, meaning we will first find $f(3)$, and then substitute that into $g(x)$.

1. First, calculate $f(3)$: $f(3) = 3^2 = 9$
2. Now, substitute $f(3) = 9$ into $g(x)$: $g(9) = 9 - 5 = 4$ So, $(g \circ f)(3) = 4$.

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Would you like further details on these steps, or do you have any questions?

Related Questions:

1. What is $(g \circ f)(x)$ for any $x$?
2. Can you expand $(x - 5)^2$ using another method?
3. What happens when $f(x)$ and $g(x)$ are different types of functions (e.g., exponential or trigonometric)?
4. How do you differentiate $(f \circ g)(x)$?
5. What are the domain and range of $(f \circ g)(x)$?

Tip:

When performing composition of functions, always be careful with the order in which you substitute functions. $f(g(x))$ is not the same as $g(f(x))$.