Complete the table of inputs and outputs for the given function.
$g(x)=3-8 x$
$x$ | $g(x)$ |
□ | 0 |
0 | □ |
□ | -5 |
3 | □ |
The problem asks us to evaluate a linear function \(g(x) = 3 - 8x\) at different points or solve for the input \(x\) when given specific outputs \(g(x)\). A function is a rule that assigns each input \(x\) exactly one output \(g(x)\). To complete the table, we will use algebraic manipulation to either solve for \(x\) given an output or evaluate the expression at a given input. The function given is \(g(x) = 3 - 8x\).
For the first row, we need to find \(x\) such that \(g(x) = 0\).
Setting the function equal to zero:
\[3 - 8x = 0\]
Subtracting 3 from both sides:
\[-8x = -3\]
Dividing by -8:
\[x = \frac{-3}{-8} = \frac{3}{8}\]
So, when \(g(x) = 0\), \(x = \frac{3}{8}\).
For the second row, we are given \(x = 0\) and need to find \(g(0)\).
Substituting \(x = 0\) into the function:
\[g(0) = 3 - 8(0)\]
\[g(0) = 3 - 0 = 3\]
So, when \(x = 0\), \(g(0) = 3\).
For the third row, we need to find \(x\) such that \(g(x) = -5\).
Setting the function equal to -5:
\[3 - 8x = -5\]
Subtracting 3 from both sides:
\[-8x = -5 - 3\]
\[-8x = -8\]
Dividing by -8:
\[x = \frac{-8}{-8} = 1\]
So, when \(g(x) = -5\), \(x = 1\).
For the fourth row, we are given \(x = 3\) and need to find \(g(3)\).
Substituting \(x = 3\) into the function:
\[g(3) = 3 - 8(3)\]
\[g(3) = 3 - 24 = -21\]
So, when \(x = 3\), \(g(3) = -21\).
Linear functions like \(g(x) = 3 - 8x\) are fundamental in modeling real-world scenarios where something changes at a constant rate. For instance, consider a pool of water initially containing 3 units of water, and you are draining it at a rate of 8 units per minute. The function \(g(x) = 3 - 8x\) models the remaining amount of water in the pool after \(x\) minutes. By using the function, you can determine exactly when the pool will be empty or how much water remains after a certain amount of time has elapsed, which is essential for resource management and planning.