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Let's solve the problem step by step.

### Problem Breakdown:
1. **Point Given**: The linear function \( f(x) \) passes through the point \( (3, 4) \).
2. **Rate of Change**: For every increase of 5 units in \( x \), \( f(x) \) increases by 3 units.

### Step 1: Determine the Slope
The slope \( m \) of the linear function is the rate of change of \( f(x) \) with respect to \( x \). Given that for every increase of 5 units in \( x \), \( f(x) \) increases by 3 units, the slope \( m \) is:
\[
m = \frac{\text{Change in } f(x)}{\text{Change in } x} = \frac{3}{5}
\]

### Step 2: Use Point-Slope Form
The point-slope form of a linear equation is:
\[
f(x) - y_1 = m(x - x_1)
\]
Substitute the point \( (3, 4) \) and the slope \( m = \frac{3}{5} \):
\[
f(x) - 4 = \frac{3}{5}(x - 3)
\]

### Step 3: Simplify the Equation
Distribute the slope on the right-hand side:
\[
f(x) - 4 = \frac{3}{5}x - \frac{9}{5}
\]
Add 4 to both sides to solve for \( f(x) \):
\[
f(x) = \frac{3}{5}x - \frac{9}{5} + 4
\]
Convert 4 to a fraction with a denominator of 5:
\[
f(x) = \frac{3}{5}x - \frac{9}{5} + \frac{20}{5}
\]
Combine the fractions:
\[
f(x) = \frac{3}{5}x + \frac{11}{5}
\]

### Step 4: Match the Equation with the Given Options
The correct answer matches option C:
\[
\boxed{f(x) = \frac{3}{5}x + \frac{11}{5}}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you find the slope when given two points?
2. What is the general formula for the point-slope form of a linear equation?
3. How do you convert a decimal to a fraction?
4. What is the importance of the slope in a linear equation?
5. How do you verify if a point lies on a given linear equation?

### Tip:
When dealing with linear functions, always start by identifying the slope and use the point-slope form to easily write the equation.

Learn how to solve a linear function problem step by step. This problem involves finding the equation of a linear function given a point and the rate of change. It covers key concepts like linear functions, point-slope form, and slope calculation. Suitable for Grades 9-12 students.

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