https://file.aiquickdraw.com/mathbot/file/172797803517045e8a22t.jpg

The problem asks to solve two tasks based on the functions \( f(x) \) and \( g(x) \).

### Given information:
- Function \( f(x) = \frac{3}{2}x - 4 \), which crosses the x-axis at \( x = 8 \).
- The graph of \( g(x) \) passes through the points \( (0, -3) \) and \( (4, 5) \).

### Part a) Find the equation of the function \( g(x) \).
To find the equation of \( g(x) \), we can use the fact that it is a straight line passing through two points, \( (0, -3) \) and \( (4, 5) \).

We can use the formula for the equation of a line in slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

1. **Calculate the slope** \( m \):
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (-3)}{4 - 0} = \frac{8}{4} = 2
   \]
   So the slope \( m = 2 \).

2. **Find the y-intercept** \( b \) by using one of the points, say \( (0, -3) \):
   \[
   y = mx + b \quad \Rightarrow \quad -3 = 2(0) + b \quad \Rightarrow \quad b = -3
   \]

Therefore, the equation of \( g(x) \) is:
\[
g(x) = 2x - 3
\]

### Part b) Find the intersection point of the graphs of \( f(x) \) and \( g(x) \).
To find the intersection point, we need to set \( f(x) = g(x) \) and solve for \( x \).

1. **Set the functions equal to each other**:
   \[
   \frac{3}{2}x - 4 = 2x - 3
   \]

2. **Solve for \( x \)**:
   First, eliminate the fractions by multiplying through by 2:
   \[
   3x - 8 = 4x - 6
   \]

   Rearrange terms:
   \[
   3x - 4x = -6 + 8 \quad \Rightarrow \quad -x = 2 \quad \Rightarrow \quad x = -2
   \]

3. **Find the y-coordinate** by plugging \( x = -2 \) into either \( f(x) \) or \( g(x) \). I'll use \( g(x) = 2x - 3 \):
   \[
   g(-2) = 2(-2) - 3 = -4 - 3 = -7
   \]

So, the intersection point is \( (-2, -7) \).

### Final Answer:
1. The equation of \( g(x) \) is \( g(x) = 2x - 3 \).
2. The intersection point of \( f(x) \) and \( g(x) \) is \( (-2, -7) \).

---

Would you like further details on any part of the solution? Here are some related questions you might find useful:

1. How can I graph \( f(x) \) and \( g(x) \)?
2. What is the significance of the slope in linear equations?
3. Can we solve part (b) using a different method, like substitution or graphical representation?
4. How do you find the intersection of two nonlinear functions?
5. How do we interpret the intersection point in terms of real-world applications?

**Tip:** When finding the slope between two points, always subtract coordinates in the same order to avoid errors.

Funktionen f(x) = (3/2)x - 4 skærer x-aksen ved x = 8. Grafen for g(x) går gennem punkterne (0, -3) og (4, 5). Bestem funktionsforskriften for g(x). Bestem skæringspunktet mellem graferne for f og g.

Solve a function problem involving linear equations. Determine the equation of g(x) passing through points (0, -3) and (4, 5), and find its intersection with f(x) = (3/2)x - 4. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation