rcise: Define the following expressions on CAS, then use the solve function to find when $$ f(x)=g(x) $$ $$ \begin{aligned} f(x)=\frac{1}{2} x+1 \ g(x)=5-2 x\end{aligned} $$

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To determine when the functions $$ f(x) $$ and $$ g(x) $$ are equal, you can use a Computer Algebra System (CAS) to define the expressions and solve for $$ x $$. Here's a step-by-step guide on how to achieve this:

### 1. Define the Functions

First, define the two functions $$ f(x) $$ and $$ g(x) $$ in your CAS environment. The definitions based on your expressions are:

$$
\begin{aligned}
f(x) &= \frac{1}{2}x + 1 \
g(x) &= 5 - 2x
\end{aligned}
$$

### 2. Set the Functions Equal to Each Other

To find the value of $$ x $$ for which $$ f(x) = g(x) $$, set the two expressions equal:

$$
\frac{1}{2}x + 1 = 5 - 2x
$$

### 3. Solve for $$ x $$ Using the `Solve` Function

Use the `solve` function to find the value of $$ x $$ that satisfies the equation. Below is how you can perform this in different CAS environments:

#### **a. Using Mathematica**

```mathematica
(* Define the functions *)
f[x_] := (1/2) x + 1
g[x_] := 5 - 2 x

(* Set f(x) equal to g(x) and solve for x *)
solution = Solve[f[x] == g[x], x]
```

**Output:**
```mathematica
{{x -> 8/5}}
```

#### **b. Using SymPy (Python Library)**

```python
from sympy import symbols, Eq, solve

# Define the variable
x = symbols('x')

# Define the functions
f_x = (1/2)*x + 1
g_x = 5 - 2*x

# Set up the equation f(x) = g(x)
equation = Eq(f_x, g_x)

# Solve for x
solution = solve(equation, x)
print(solution)
```

**Output:**
```
[8/5]
```

#### **c. Using TI Calculators (e.g., TI-89)**

1. **Define the Functions:**
   - Press the `Y=` button.
   - Enter `0.5x + 1` for $$ Y_1 $$.
   - Enter `5 - 2x` for $$ Y_2 $$.

2. **Find the Intersection:**
   - Press the `2nd` button, then `Calc` (above the `Trace` button).
   - Choose `5: intersect`.
   - Select the two functions you've defined.
   - Press `Enter` to get the solution.

**Output:**
```
x = 8/5
```

### 4. Interpretation of the Solution

The solution $$ x = \frac{8}{5} $$ (or $$ x = 1.6 $$) indicates the point where $$ f(x) $$ and $$ g(x) $$ intersect. To confirm, you can substitute $$ x = \frac{8}{5} $$ back into both functions:

$$
\begin{aligned}
f\left(\frac{8}{5}\right) &= \frac{1}{2} \left(\frac{8}{5}\right) + 1 = \frac{4}{5} + 1 = \frac{9}{5} \
g\left(\frac{8}{5}\right) &= 5 - 2 \left(\frac{8}{5}\right) = 5 - \frac{16}{5} = \frac{25}{5} - \frac{16}{5} = \frac{9}{5}
\end{aligned}
$$

Both functions evaluate to $$ \frac{9}{5} $$ when $$ x = \frac{8}{5} $$, confirming the solution.

### 5. Graphical Interpretation

Plotting both functions on the same graph will show that they intersect at the point $$ \left(\frac{8}{5}, \frac{9}{5}\right) $$.

---

**Summary:** By defining the functions $$ f(x) = \frac{1}{2}x + 1 $$ and $$ g(x) = 5 - 2x $$ in a CAS and using the `solve` function to set them equal, we find that $$ f(x) = g(x) $$ when $$ x = \frac{8}{5} $$.
To find when $$ f(x) = g(x) $$, set $$ \frac{1}{2}x + 1 = 5 - 2x $$ and solve for $$ x $$. The solution is $$ x = \frac{8}{5} $$.

rcise: Define the following expressions on CAS, then use the solve function to find when \( f(x)=g(x) \) \[ \begin{aligned} f(x)=\frac{1}{2} x+1 \\ g(x)=5-2 x\end{aligned} \]

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