To solve the given problems, we'll perform the composition of functions $(f \circ g)(x)$, which is defined as $f(g(x))$. This means we substitute $g(x)$ into $f(x)$ wherever $x$ appears.

Let's work through each of the assigned problems.

Problem 1:

Given:

• $f(x) = x + 1$
• $g(x) = 2x - 3$

Find $(f \circ g)(x)$:

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

Substitute $g(x)$ into $f(x)$:

$f(2x - 3) = (2x - 3) + 1 = 2x - 2$

Answer: $(f \circ g)(x) = 2x - 2$

Problem 3:

Given:

• $f(x) = x^2 + 1$
• $g(x) = x - 3$

Find $(f \circ g)(x)$:

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

Substitute $g(x)$ into $f(x)$:

$f(x - 3) = (x - 3)^2 + 1$

Expand $(x - 3)^2$:

$(x - 3)^2 = x^2 - 6x + 9$

So,

$f(x - 3) = x^2 - 6x + 9 + 1 = x^2 - 6x + 10$

Answer: $(f \circ g)(x) = x^2 - 6x + 10$

Problem 5:

Given:

• $f(x) = x - 2$
• $g(x) = \frac{1}{x}$

Find $(f \circ g)(x)$:

$(f \circ g)(x) = f(g(x)) = f\left(\frac{1}{x}\right)$

Substitute $g(x)$ into $f(x)$:

$f\left(\frac{1}{x}\right) = \frac{1}{x} - 2$

Answer: $(f \circ g)(x) = \frac{1}{x} - 2$

Problem 7:

Given:

• $f(x) = 3x^2 + 1$
• $g(x) = x - 3$

Find $(f \circ g)(x)$:

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

Substitute $g(x)$ into $f(x)$:

$f(x - 3) = 3(x - 3)^2 + 1$

Expand $(x - 3)^2$:

$(x - 3)^2 = x^2 - 6x + 9$

So,

$f(x - 3) = 3(x^2 - 6x + 9) + 1 = 3x^2 - 18x + 27 + 1 = 3x^2 - 18x + 28$

Answer: $(f \circ g)(x) = 3x^2 - 18x + 28$

Problem 9:

Given:

• $f(x) = \frac{2}{x}$
• $g(x) = \frac{1}{x}$

Find $(f \circ g)(x)$:

$(f \circ g)(x) = f(g(x)) = f\left(\frac{1}{x}\right)$

Substitute $g(x)$ into $f(x)$:

$f\left(\frac{1}{x}\right) = \frac{2}{\frac{1}{x}} = 2x$

Answer: $(f \circ g)(x) = 2x$

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Would you like a detailed explanation for any of these problems, or do you have any questions?

Here are some related questions for further practice:

1. What is the difference between $f(g(x))$ and $g(f(x))$?
2. How do you find the domain of a composite function $(f \circ g)(x)$?
3. Can a composite function $(f \circ g)(x)$ be equal to $(g \circ f)(x)$? Provide an example.
4. How do you determine if two functions are inverses of each other using composition?
5. What is the significance of composition in function transformations?
6. How does the order of function composition affect the result?
7. Solve $(f \circ g)(x)$ if $f(x) = \sqrt{x}$ and $g(x) = x^2 + 2$.
8. How do you interpret the composition of functions graphically?

Tip: When composing functions, carefully track the order of substitution to avoid mistakes. The composition $(f \circ g)(x)$ is not the same as $(g \circ f)(x)$.