If f(x) = 1/x and g(x) = 3x + 2, find (g o g)(-4).
Understand the Problem
The question is asking to find the value of (g o g)(-4), where g(x) is defined as 3x + 2/x. This involves evaluating the function g at -4 and then using that result as the input for g again.
Answer
The answer to $(g \circ g)(-4)$ is $\frac{19}{5}$.
Answer for screen readers
The final answer is $\frac{19}{5}$.
Steps to Solve
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Evaluate g(-4)
First, we need to find the value of $g(-4)$. The function is defined as: $$ g(x) = \frac{3x + 2}{x} $$ Substituting $x = -4$ into the function: $$ g(-4) = \frac{3(-4) + 2}{-4} = \frac{-12 + 2}{-4} = \frac{-10}{-4} = \frac{10}{4} = \frac{5}{2} $$ -
Evaluate g(g(-4))
Now we need to substitute $g(-4)$ back into the function $g$. So we need to evaluate $g\left(\frac{5}{2}\right)$: $$ g\left(\frac{5}{2}\right) = \frac{3 \left(\frac{5}{2}\right) + 2}{\frac{5}{2}} = \frac{\frac{15}{2} + 2}{\frac{5}{2}} $$ We need to convert 2 into a fraction: $$ 2 = \frac{4}{2} $$ Then we continue: $$ g\left(\frac{5}{2}\right) = \frac{\frac{15}{2} + \frac{4}{2}}{\frac{5}{2}} = \frac{\frac{19}{2}}{\frac{5}{2}} $$ To divide by a fraction, we multiply by its reciprocal: $$ g\left(\frac{5}{2}\right) = \frac{19}{2} \times \frac{2}{5} = \frac{19 \cdot 2}{2 \cdot 5} = \frac{19}{5} $$ -
Final Result
Thus, the final value of $(g \circ g)(-4)$ is: $$ g(g(-4)) = \frac{19}{5} $$
The final answer is $\frac{19}{5}$.
More Information
This problem uses function composition, which is a fundamental concept in algebra. The function $g(x)$ combines several operations like multiplication and division. It's essential to evaluate inside out when dealing with such compositions.
Tips
- Incorrect order of operations: It's crucial to evaluate $g(-4)$ first before applying it to $g$ again.
- Not converting integers to fractions adequately: Failing to convert numbers properly can lead to mistakes in fraction manipulation.
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