- Calculate \(f(x) = -\left(\frac{1}{2}\right)^x - 1\) for each input \(x\) of the provided ordered pairs.
- \((0, -2)\): \(f(0) = -1 - 1 = -2\), so it is \(\boxed{\text{On the graph}}\).
- \((-1, -3)\): \(f(-1) = -2 - 1 = -3\), so it is \(\boxed{\text{On the graph}}\).
- \((1, -\frac{1}{2})\): \(f(1) = -0.5 - 1 = -1.5\), so it is \(\boxed{\text{Not on the graph}}\).
- \((-3, -9)\): \(f(-3) = -8 - 1 = -9\), so it is \(\boxed{\text{On the graph}}\).
Explanation
- Understanding the Problem and Function
The function provided is \(f(x) = -\left(\frac{1}{2}\right)^x - 1\). To determine if an ordered pair \((x_0, y_0)\) lies on the graph of this function, we must verify if substituting \(x_0\) into the function results in \(y_0\), that is, if \(f(x_0) = y_0\). We will examine each pair one by one.
- Checking (0, -2)
For the pair \((0, -2)\), we substitute \(x = 0\) into the function:
\[f(0) = -\left(\frac{1}{2}\right)^0 - 1 = -(1) - 1 = -2.\]
Since \(f(0) = -2\), the point \((0, -2)\) satisfies the equation and is on the graph.
- Checking (-1, -3)
For the pair \((-1, -3)\), we substitute \(x = -1\) into the function:
\[f(-1) = -\left(\frac{1}{2}\right)^{-1} - 1 = -(2) - 1 = -3.\]
Since \(f(-1) = -3\), the point \((-1, -3)\) satisfies the equation and is on the graph.
- Checking (1, -1/2)
For the pair \((1, -\frac{1}{2})\), we substitute \(x = 1\) into the function:
\[f(1) = -\left(\frac{1}{2}\right)^1 - 1 = -0.5 - 1 = -1.5 = -\frac{3}{2}.\]
Since \(f(1) = -\frac{3}{2}
eq -\frac{1}{2}\), the point \((1, -\frac{1}{2})\) does not satisfy the equation and is not on the graph.
- Checking (-3, -9)
For the pair \((-3, -9)\), we substitute \(x = -3\) into the function:
\[f(-3) = -\left(\frac{1}{2}\right)^{-3} - 1 = -(2^3) - 1 = -8 - 1 = -9.\]
Since \(f(-3) = -9\), the point \((-3, -9)\) satisfies the equation and is on the graph.
- Conclusion
Based on our calculations, the results are:
- \((0, -2)\): On the graph
- \((-1, -3)\): On the graph
- \((1, -\frac{1}{2})\): Not on the graph
- \((-3, -9)\): On the graph
This completes our analysis of the given points.
Examples
Functions are used extensively in economics to model demand and supply. For example, if the demand for a product follows a specific functional relationship with its price, say \(D(p) = 500 \cdot e^{-0.1p}\), economists can check whether specific market price-demand observations lie on this "demand curve." If an observation deviates significantly from the calculated curve, it may indicate that external market factors are influencing the demand, allowing analysts to adjust their models for better predictions.