The table shows values for functions $f(x)$ and $g(x)$.

| $x$ | $f(x)=2^x-4$ | $g(x)=\frac{3}{2}x-3$ |

| :---: | :---: | :---: |

| -1 | $-\frac{7}{2}$ | $-\frac{9}{2}$ |

| 0 | $-3$ | $-3$ |

| 1 | $-2$ | $-\frac{3}{2}$ |

| 2 | $0$ | $0$ |

| 3 | $4$ | $\frac{3}{2}$ |

| 4 | $12$ | $3$ |

| 5 | $28$ | $\frac{9}{2}$ |

What is the solution to $f(x)=g(x)$?

We are given two functions, $f(x) = 2^x - 4$ and $g(x) = \frac{3}{2}x - 3$, along with a table of their values for various integers $x$ in the range $-1$ to $5$. Our task is to identify the values of $x$ for which these two functions are equal, meaning $f(x) = g(x)$. To find these points, we compare the values of both functions side-by-side from the provided data.

Identifying Solutions from Data

By observing the table, we look for rows where the output of $f(x)$ is identical to the output of $g(x)$.

For $x = -1$: $f(-1) = -3.5$ and $g(-1) = -4.5$. Not equal.
For $x = 0$: $f(0) = -3$ and $g(0) = -3$. They are equal! Thus, $x = 0$ is a solution.
For $x = 1$: $f(1) = -2$ and $g(1) = -1.5$. Not equal.
For $x = 2$: $f(2) = 0$ and $g(2) = 0$. They are equal! Thus, $x = 2$ is a solution.
For $x = 3, 4, 5$: The values diverge and are not equal.

Based on the table, the solutions for $f(x) = g(x)$ are $x = 0$ and $x = 2$.

Verification and Conclusion

To ensure our findings are correct, we substitute the values back into the equations:
For $x = 0$:
$f(0) = 2^0 - 4 = 1 - 4 = -3$
$g(0) = \frac{3}{2}(0) - 3 = -3$
Since $f(0) = g(0)$, the solution is valid.

For $x = 2$:
$f(2) = 2^2 - 4 = 4 - 4 = 0$
$g(2) = \frac{3}{2}(2) - 3 = 3 - 3 = 0$
Since $f(2) = g(2)$, the solution is valid.
We have confirmed that the solutions are $x = 0$ and $x = 2$, represented as $\boxed{0, 2}$.

Examples

Imagine you are managing two different investment accounts. Account A grows exponentially, modeled by a function like $f(x) = 2^x - 4$, while Account B grows linearly, modeled by $g(x) = \frac{3}{2}x - 3$. By identifying the points where these two functions intersect, you can determine exactly when the balances in both accounts are equal, helping you make informed decisions about your financial strategies and future planning. This method is fundamental in various fields, from business and finance to engineering and physics, to understand where two different systems behave identically.