Questions: For the functions f(x) = 2 / (x + 3) and g(x) = 5 / (x - 1), find the composition f · g and simplify your answer as much as possible. Write the domain using interval notation. (f · g)(x) = Domain of f · g :

Solution Steps

To find the composition \( (f \cdot g)(x) \), we need to substitute \( g(x) \) into \( f(x) \). Then, we simplify the resulting expression. Finally, we determine the domain by identifying the values of \( x \) that make the denominator zero in either \( f(x) \) or \( g(x) \).

The composition of two functions \( f \) and \( g \), denoted as \( (f \cdot g)(x) \), is defined as: \[ (f \cdot g)(x) = f(x) \cdot g(x) \]

Given: \[ f(x) = \frac{2}{x+3} \quad \text{and} \quad g(x) = \frac{5}{x-1} \] Substitute these into the composition: \[ (f \cdot g)(x) = \left( \frac{2}{x+3} \right) \cdot \left( \frac{5}{x-1} \right) \]

Multiply the fractions: \[ (f \cdot g)(x) = \frac{2 \cdot 5}{(x+3)(x-1)} = \frac{10}{(x+3)(x-1)} \]

The domain of \( f \cdot g \) is the set of all \( x \) values for which the expression is defined. The expression \( \frac{10}{(x+3)(x-1)} \) is undefined when the denominator is zero. Therefore, we need to find the values of \( x \) that make the denominator zero: \[ (x+3)(x-1) = 0 \] Solving for \( x \): \[ x+3 = 0 \quad \Rightarrow \quad x = -3 \] \[ x-1 = 0 \quad \Rightarrow \quad x = 1 \] Thus, the function is undefined at \( x = -3 \) and \( x = 1 \).

Final Answer