Question
rcise: Define the following expressions on CAS, then use the solve function to find when \( f(x)=g(x) \) \[ \begin{aligned} f(x)=\frac{1}{2} x+1 \\ g(x)=5-2 x\end{aligned} \]
Ask by Floyd Moran. in Australia
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
To find when \( f(x) = g(x) \), set \( \frac{1}{2}x + 1 = 5 - 2x \) and solve for \( x \). The solution is \( x = \frac{8}{5} \).
Solution
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 \( x = \frac{8}{5} \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To define the functions on a Computer Algebra System (CAS), you would typically input them like this: ``` f(x) := (1/2)*x + 1 g(x) := 5 - 2*x ``` Next, to find the intersection point where \( f(x) = g(x) \), you would set the two functions equal to each other and utilize the solve function: ``` solve(f(x) = g(x), x) ``` When you execute this, the CAS will give you the value of \( x \) where the two functions intersect, which you can then substitute back into either of the equations to find the corresponding \( y \) value! It’s like finding the hidden treasure where two paths meet!
