$x$ | $3 x$ | $y=3 x-5$ |
-3 | □ | □ |
-2 | ||
-1 | ||
0 | ||
1 | ||
2 | ||
3 | ||
The problem requires us to evaluate two linear functions, \(f(x) = 3x\) and \(g(x) = 3x - 5\), for a set of integer values of \(x\) ranging from \(-3\) to \(3\). By substituting each value of \(x\) into these expressions, we can determine the corresponding outputs and complete the table.
To find the values for the column \(3x\), we multiply each input \(x\) by \(3\). Similarly, to find the values for the column \(3x-5\), we multiply each input \(x\) by \(3\) and then subtract \(5\).
For \(x = -3\):
\(f(-3) = 3(-3) = -9\)
\(g(-3) = 3(-3) - 5 = -9 - 5 = -14\)
For \(x = -2\):
\(f(-2) = 3(-2) = -6\)
\(g(-2) = 3(-2) - 5 = -6 - 5 = -11\)
For \(x = -1\):
\(f(-1) = 3(-1) = -3\)
\(g(-1) = 3(-1) - 5 = -3 - 5 = -8\)
For \(x = 0\):
\(f(0) = 3(0) = 0\)
\(g(0) = 3(0) - 5 = -5\)
For \(x = 1\):
\(f(1) = 3(1) = 3\)
\(g(1) = 3(1) - 5 = 3 - 5 = -2\)
For \(x = 2\):
\(f(2) = 3(2) = 6\)
\(g(2) = 3(2) - 5 = 6 - 5 = 1\)
For \(x = 3\):
\(f(3) = 3(3) = 9\)
\(g(3) = 3(3) - 5 = 9 - 5 = 4\)
Collecting the results from the calculations, we can fill the table as follows:
| \(x\) | \(3x\) | \(3x - 5\) \) \(-3\) | \(-9\) | \(-14\) \) \(-2\) | \(-6\) | \(-11\) \) \(-1\) | \(-3\) | \(-8\) \) \(0\) | \(0\) | \(-5\) \) \(1\) | \(3\) | \(-2\) \) \(2\) | \(6\) | \(1\) \) \(3\) | \(9\) | \(4\) \) |
Linear functions are everywhere in daily life! Consider a simple pricing model where you pay an initial fee and then a per-item cost. For example, if you are renting equipment, you might pay a flat service fee of \(\$5\) and then \(\$3\) for every hour you use it. Your total cost \(y\) in dollars for \(x\) hours of use can be represented as \(y = 3x + 5\) (or here, \(-5\) if you think of a discount). Tracking these values in a table helps you plan your budget effectively and understand how your total cost changes as you consume more of a service.