- Quadratic functions \(f(x) = -(x+1)^2 - 4\) and \(f(x) = x^2 + 7x - 9\) have bounded ranges.
- The exponential function \(f(x) = 2^{x+3}\) is restricted to positive values \((0, \infty)\).
- Linear functions \(f(x) = -4x + 11\) and \(f(x) = \frac{2}{3}x - 8\) with non-zero slopes have a range of \((-\infty, \infty)\).
- The correct functions are: \(\boxed{f(x) = -4x + 11, f(x) = \frac{2}{3}x - 8}\).
Explanation
- Understanding the Range of Functions
To determine which functions have a range of all real numbers, we need to understand the behavior of each function type. The range of a function is the set of all possible output values (\(y\)-values) it can produce. A function has a range of all real numbers \((-\infty, \infty)\) if it can take any real value. We will analyze each of the five given functions.
- Analyzing Each Function
- \(f(x) = -(x+1)^2 - 4\): This is a quadratic function of the form \(f(x) = a(x-h)^2 + k\), representing a downward-opening parabola with its vertex at \((-1, -4)\). Its maximum value is \(-4\), so the range is \((-\infty, -4]\). This is not all real numbers.
- \(f(x) = 2^{x+3}\): This is an exponential function. Exponential functions of the form \(b^{x+k}\) (where \(b>0, b
eq 1\)) are always positive, producing outputs in the interval \((0, \infty)\). This is not all real numbers.
- \(f(x) = -4x + 11\): This is a linear function in the form \(f(x) = mx + b\). Since the slope \(m = -4\) is non-zero, the function is strictly decreasing. As \(x\) approaches \(\infty\), \(f(x)\) approaches \(-\infty\), and as \(x\) approaches \(-\infty\), \(f(x)\) approaches \(\infty\). Therefore, the range is \((-\infty, \infty)\).
- \(f(x) = \frac{2}{3}x - 8\): This is also a linear function in the form \(f(x) = mx + b\). The slope \(m = \frac{2}{3}\) is non-zero, making the function strictly increasing. Consequently, it covers all real values, so the range is \((-\infty, \infty)\).
- \(f(x) = x^2 + 7x - 9\): This is a quadratic function that opens upward since the coefficient of \(x^2\) is positive. It has a minimum value at its vertex. Therefore, its range is \([k, \infty)\) where \(k\) is the y-coordinate of the vertex. This is not all real numbers.
- Conclusion
Based on our analysis, the linear functions \(f(x) = -4x + 11\) and \(f(x) = \frac{2}{3}x - 8\) cover all real numbers. Quadratic functions and exponential functions have bounded or restricted ranges and therefore do not have a range of all real numbers. The correct functions are \(\boxed{f(x) = -4x + 11}\) and \(\boxed{f(x) = \frac{2}{3}x - 8}\).
Examples
In real-world applications, identifying the range of a function is crucial for understanding the limits of a system. For example, if a function models the cost of production, the range tells us the minimum and maximum possible costs. A linear cost function \(f(x) = 5x + 100\) (where \(x\) is the number of units) has a range of \([100, \infty)\) because you cannot produce negative items or have negative costs. In contrast, temperature changes modeled by a linear function can span across negative, zero, and positive values, indicating a range of \((-\infty, \infty)\) within a certain physical context. Knowing this allows scientists and engineers to predict and set boundaries for their models.