- Test each table by plugging in the x-values to see if they produce the corresponding y-values for the given functions.
- Identify that Table 4 represents the function \(f(x) = 4x - 3\).
- Identify that Table 2 represents the function \(g(x) = \frac{1}{4}x + \frac{3}{4}\).
- Observe that the set of \((x, y)\) pairs in Table 4 are the \((y, x)\) pairs of Table 2, confirming they are inverses.
- The two correct tables are the second and fourth options.
Explanation
- Understanding the Inverse Relationship
To determine which tables verify that the functions \(f(x) = 4x - 3\) and \(g(x) = \frac{1}{4}x + \frac{3}{4}\) are inverses, we first recall the definition of inverse functions. Two functions \(f\) and \(g\) are inverses if for every pair \((a, b)\) that satisfies \(b = f(a)\), the pair \((b, a)\) satisfies \(a = g(b)\).
We analyze the given tables by testing their \((x, y)\) pairs against the formulas for \(f(x)\) and \(g(x)\). A table represents the function \(f\) if every entry satisfies \(y = 4x - 3\), and it represents the function \(g\) if every entry satisfies \(y = \frac{1}{4}x + \frac{3}{4}\).
- Testing the Tables
We test each table:
- Table 1: \(x = [1, 3, 5, 7, 9], y = [1, 3, 5, 7, 9]\). Checking \(y = 4x - 3\): for \(x=1\), \(y=1
eq 4(1)-3=1\). Actually, \(4(1)-3=1\), \(4(3)-3=9
eq 3\). So Table 1 is neither.
- Table 2: \(x = [-23, -15, -3, 1, 13], y = [-5, -3, 0, 1, 4]\). Checking \(y = g(x) = \frac{1}{4}x + \frac{3}{4}\):
- For \(x=-23\), \(y = \frac{-23}{4} + \frac{3}{4} = \frac{-20}{4} = -5\). (Correct)
- For \(x=-15\), \(y = \frac{-15}{4} + \frac{3}{4} = \frac{-12}{4} = -3\). (Correct)
- For \(x=-3\), \(y = \frac{-3}{4} + \frac{3}{4} = 0\). (Correct)
- For \(x=1\), \(y = \frac{1}{4} + \frac{3}{4} = 1\). (Correct)
- For \(x=13\), \(y = \frac{13}{4} + \frac{3}{4} = 4\). (Correct)
Table 2 represents \(g(x) = \frac{1}{4}x + \frac{3}{4}\).
- Table 4: \(x = [-5, -3, 0, 1, 4], y = [-23, -15, -3, 1, 13]\). Checking \(y = f(x) = 4x - 3\):
- For \(x=-5\), \(y = 4(-5) - 3 = -23\). (Correct)
- For \(x=-3\), \(y = 4(-3) - 3 = -15\). (Correct)
- For \(x=0\), \(y = 4(0) - 3 = -3\). (Correct)
- For \(x=1\), \(y = 4(1) - 3 = 1\). (Correct)
- For \(x=4\), \(y = 4(4) - 3 = 13\). (Correct)
Table 4 represents \(f(x) = 4x - 3\).
- Verifying the Inverse Property and Conclusion
Since Table 4 represents \(f(x) = 4x - 3\) and Table 2 represents \(g(x) = \frac{1}{4}x + \frac{3}{4}\), we verify if they confirm the inverse property. The set of points in Table 4 is \(\{(-5, -23), (-3, -15), (0, -3), (1, 1), (4, 13)\}\). Swapping the coordinates of these points gives \(\{(-23, -5), (-15, -3), (-3, 0), (1, 1), (13, 4)\}\). These swapped points are precisely the points given in Table 2. Therefore, Table 2 and Table 4 are the ones that can be used to verify that the functions are inverses.
Examples
In real life, inverse functions are used in currency conversion. If a function \(f(x)\) converts US dollars to another currency (e.g., Euros) based on an exchange rate, its inverse function \(f^{-1}(y)\) allows you to calculate back how many US dollars you would get for a given amount of Euros. By looking at tables of these conversions, you can easily verify the relationship between these two currencies and ensure that the conversion back and forth is accurate.