- Analyze the range of \(f(x) = \frac{1}{x}\) for all \(x
eq 0\), finding that it covers all nonzero real numbers.
- Analyze the range of \(g(x) = \frac{6}{x}\) for all \(x
eq 0\), finding that it also covers all nonzero real numbers.
- Compare the two ranges to observe that they are identical.
- Final Conclusion: The range of both \(f(x)\) and \(g(x)\) is all nonzero real numbers.
Explanation
- Analyzing the Range of the Parent Function \(f(x)\)
The parent function is given by \(f(x) = \frac{1}{x}\), where \(x
eq 0\). As \(x\) takes on values arbitrarily close to \(0\) from the positive side, \(f(x)\) approaches \(+\infty\), and as \(x\) becomes very large, \(f(x)\) approaches \(0\). Similarly, as \(x\) takes on values arbitrarily close to \(0\) from the negative side, \(f(x)\) approaches \(-\infty\), and as \(x\) becomes very large in the negative direction, \(f(x)\) approaches \(0\). Thus, the function \(f(x)\) can take any real value except \(0\). Therefore, the range of \(f(x)\) is all nonzero real numbers.
- Analyzing the Range of the Transformed Function \(g(x)\)
The function \(g(x) = \frac{6}{x}\) is defined for all \(x
eq 0\). Similar to the parent function, as \(x\) varies over all nonzero real numbers, the value of \(g(x) = \frac{6}{x}\) covers all real numbers except \(0\). Specifically, as \(x\) approaches \(0\) from the positive side, \(g(x)\) approaches \(+\infty\); as \(x\) approaches \(+\infty\), \(g(x)\) approaches \(0\). As \(x\) approaches \(0\) from the negative side, \(g(x)\) approaches \(-\infty\); as \(x\) approaches \(-\infty\), \(g(x)\) approaches \(0\). No matter how large or small \(x\) is, \(g(x)\) will never equal \(0\). Therefore, the range of \(g(x)\) is also all nonzero real numbers.
- Comparison of Ranges and Conclusion
By comparing our findings, we see that the set of all possible output values for \(f(x) = \frac{1}{x}\) is the set of all nonzero real numbers. Similarly, the set of all possible output values for \(g(x) = \frac{6}{x}\) is also the set of all nonzero real numbers. Both functions share the same range. Thus, the correct conclusion is that the range of both \(f(x)\) and \(g(x)\) is all nonzero real numbers.
Examples
In real-life scenarios, the concept of a range helps us understand the limitations of a system. For instance, consider the intensity of a light source at a distance \(d\). The light intensity \(I\) often follows an inverse relationship \(I = \frac{k}{d^2}\). Regardless of the constant \(k\), the range of possible intensity values for all positive distances \(d\) will always be all positive real numbers. Knowing the range of a function helps scientists and engineers predict the possible output values of a system without needing to test every single input case.