1. Find $f(g(x))$ when $f(x) = 3x^2 + 2$ and $g(x) = \sqrt{1 - x}$

Step 1: Substitute $g(x) = \sqrt{1 - x}$ into $f(x)$

We are asked to find $f(g(x))$, which means we need to plug the expression for $g(x)$ into the function $f(x)$.

The function $f(x)$ is: $f(x) = 3x^2 + 2$ Now, substitute $g(x) = \sqrt{1 - x}$ into $f(x)$: $f(g(x)) = 3(\sqrt{1 - x})^2 + 2$

Step 2: Simplify

Since $(\sqrt{1 - x})^2 = 1 - x$, we have: $f(g(x)) = 3(1 - x) + 2$ Simplify further: $f(g(x)) = 3 - 3x + 2$ $f(g(x)) = 5 - 3x$

Thus, $f(g(x)) = 5 - 3x$.

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2. Find $f(g(1))$ and $g(f(1))$ given $f(x) = 5x^2 - 3x + 2$ and $g(x) = x + 3$

Part (a) Find $f(g(1))$

Step 1: Compute $g(1)$

We need to first find $g(1)$. Since $g(x) = x + 3$, substitute $x = 1$: $g(1) = 1 + 3 = 4$

Step 2: Compute $f(g(1)) = f(4)$

Now we substitute $g(1) = 4$ into $f(x)$. The function $f(x) = 5x^2 - 3x + 2$, so: $f(4) = 5(4)^2 - 3(4) + 2$ Simplify: $f(4) = 5(16) - 12 + 2 = 80 - 12 + 2 = 70$

Thus, $f(g(1)) = 70$.

Part (b) Find $g(f(1))$

Step 1: Compute $f(1)$

We need to first find $f(1)$. Since $f(x) = 5x^2 - 3x + 2$, substitute $x = 1$: $f(1) = 5(1)^2 - 3(1) + 2 = 5 - 3 + 2 = 4$

Step 2: Compute $g(f(1)) = g(4)$

Now substitute $f(1) = 4$ into $g(x)$. Since $g(x) = x + 3$, we have: $g(4) = 4 + 3 = 7$

Thus, $g(f(1)) = 7$.

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3. Is $f(g(x)) = g(f(x))$ for two distinct functions?

In general, $f(g(x))$ is not equal to $g(f(x))$ for two distinct functions. This is because composition of functions is generally not commutative.

To justify this, let’s look at an example with two distinct functions:

• Let $f(x) = 2x$ and $g(x) = x + 1$.

Now, compute both compositions:

• $f(g(x)) = 2(g(x)) = 2(x + 1) = 2x + 2$,
• $g(f(x)) = g(2x) = 2x + 1$.

Clearly, $f(g(x)) = 2x + 2$ and $g(f(x)) = 2x + 1$, so $f(g(x)) \neq g(f(x))$.

Thus, $f(g(x)) \neq g(f(x))$ in general.

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Summary of Results:

1. $f(g(x)) = 5 - 3x$
2. $f(g(1)) = 70$ and $g(f(1)) = 7$
3. In general, $f(g(x)) \neq g(f(x))$, because function composition is not commutative.

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Do you want further explanations or have any other questions?

Related Questions:

1. How do you find $g(f(g(x)))$ for given functions?
2. What is the domain of $f(g(x))$ when $f(x) = \frac{1}{x}$ and $g(x) = \sqrt{x}$?
3. What is the general property of function composition?
4. If $f(x) = x^2$ and $g(x) = x + 2$, what is $g(g(x))$?
5. Is $f(f(x))$ always equal to $f(x)^2$?

Tip:

Function composition is not commutative. Always check the order in which functions are composed.