Identifying The Characteristics of Linear and Exponential FunctionsFunctionType and Behavior f(x)=34x-230


To identify the characteristics of the function \( f(x) = 34x - 230 \), let's determine whether it is a linear or exponential function and analyze its behavior.

### 1. **Type of Function:**
   
The given function is in the form \( f(x) = 34x - 230 \). This is a linear function because it can be written in the standard form of a linear function, which is \( f(x) = mx + b \), where:

- \( m \) represents the slope of the line.
- \( b \) represents the y-intercept.

Here, \( m = 34 \) and \( b = -230 \).

### 2. **Characteristics of a Linear Function:**

- **Slope (m):** The slope of the line is 34. This means for every unit increase in \( x \), the value of \( f(x) \) increases by 34 units. A positive slope indicates the function is increasing.
  
- **Y-Intercept (b):** The y-intercept is -230. This is the point where the line crosses the y-axis, i.e., when \( x = 0 \), \( f(x) = -230 \).

- **Graph:** The graph of this function is a straight line with a positive slope, meaning it will ascend from left to right.

### 3. **Behavior of the Function:**

- **Increasing Behavior:** Since the slope is positive (34), the function is increasing for all values of \( x \). In other words, as \( x \) increases, \( f(x) \) also increases.

- **No Maximum or Minimum:** A linear function does not have a maximum or minimum value because it increases or decreases indefinitely in both directions.

- **No Curvature:** The graph of a linear function is a straight line, so there is no curvature.

### 4. **Domain and Range:**

- **Domain:** The domain of a linear function is all real numbers, \( \mathbb{R} \). In other words, \( x \) can take any real value.

- **Range:** Similarly, the range is also all real numbers, \( \mathbb{R} \), because as \( x \) varies, \( f(x) \) can take any real value.

### 5. **Intercepts:**

- **Y-Intercept:** As calculated, the y-intercept is \( -230 \), which is the value of \( f(x) \) when \( x = 0 \).

- **X-Intercept:** To find the x-intercept (where \( f(x) = 0 \)), set the function equal to zero:

  \[
  34x - 230 = 0
  \]

  Solving for \( x \):

  \[
  34x = 230 \implies x = \frac{230}{34} \approx 6.76
  \]

  So, the x-intercept is approximately 6.76.

### Summary:

- **Function Type:** Linear
- **Slope:** 34 (indicates increasing function)
- **Y-Intercept:** -230
- **X-Intercept:** Approximately 6.76
- **Domain and Range:** Both are all real numbers, \( \mathbb{R} \).

Would you like more details or have any questions?

### Related Questions:

1. What are the characteristics of a function with a negative slope?
2. How do you find the slope of a line given two points on the line?
3. How does changing the value of \( b \) affect the graph of a linear function?
4. Can a linear function have more than one x-intercept? Why or why not?
5. How do you distinguish between a linear and an exponential function?

### Tip: 
Remember, for linear functions, the graph is always a straight line, and the slope tells you whether the function is increasing (positive slope) or decreasing (negative slope).

Learn how to identify the characteristics of the linear function f(x) = 34x - 230. This function analysis covers its type, behavior, slope, y-intercept, x-intercept, domain, and range. Suitable for high school students studying linear functions.

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