Question

In Exercises \(63-66,\) (a) graph \(f \circ g\) and \(g \circ f\) and make a conjecture about the domain and range of each function. (b) Then confirm your conjectures by finding formulas for \(f \circ g\) and \(g \circ f\) . $$f(x)=\frac{2 x-1}{x+3}, g(x)=\frac{3 x+1}{2-x}$$

Short answer

The formulas for the composition of functions are \(f(g(x)) = \frac{4x +3}{6}\) and \(g(f(x)) = \frac{6x +3}{2x +5}\). The conjectured domain for \(f \circ g\) is all real numbers except 3, and for \(g \circ f\) is all real numbers except -3. Both functions can output any real number, so their range is all real numbers.

Step-by-step solution

Compute \(f \circ g\)(x)

This means we need to substitute \(g(x)\) into \(f(x)\). So, \(f(g(x)) = f(\frac{3 x+1}{2-x}) = \frac{2(\frac{3 x+1}{2-x})-1}{\frac{3 x+1}{2-x} +3}\). Simplify this expression.

Simplify \(f \circ g\)(x)

To simplify, find the common denominators and combine the fractions. The result is \(f(g(x)) = \frac{(6x +2 - 2x +1)}{(3x +1 + 6 - 3x)} = \frac{4x +3}{6}\).

Compute \(g \circ f\)(x)

Similar to step 1, substitute \(f(x)\) into \(g(x)\). So, \(g(f(x)) = g(\frac{2x -1}{x +3}) = \frac{3 (\frac{2x -1}{x +3})+1}{2 - \frac{2x -1}{x +3}}\).

Simplify \(g \circ f\)(x)

To simplify, find the common denominators and combine the fractions. The result after simplifying the expression is \(g(f(x)) = \frac{6x +3}{2x +5}\).

Make Conjectures on the Domain and Range

For the domain of \(f \circ g\) and \(g \circ f\), we should exclude values that would lead to division by zero in the fractions. So, from the expressions found for \(f \circ g\) and \(g \circ f\), we infer that the domain for both functions is all real numbers except 3 and -3, respectively. As for the range, these functions can output any real number.