Sketch the graph of the linear function, then give its domain and range. \( f(x)=5 x+1 \)

To sketch the graph of the linear function \( f(x) = 5x + 1 \), we can follow these steps: 1. **Identify the slope and y-intercept**: - The slope (m) is 5, which means for every unit increase in \( x \), \( f(x) \) increases by 5 units. - The y-intercept (b) is 1, which means the graph crosses the y-axis at the point (0, 1). 2. **Plot the y-intercept**: - Start by plotting the point (0, 1) on the graph. 3. **Use the slope to find another point**: - From (0, 1), move up 5 units (because of the slope) and 1 unit to the right to find another point. This gives us the point (1, 6). 4. **Draw the line**: - Draw a straight line through the points (0, 1) and (1, 6). Since this is a linear function, the line will continue infinitely in both directions. ### Domain and Range - **Domain**: The domain of a linear function is all real numbers, since you can input any real number for \( x \). Therefore, the domain is: \[ \text{Domain: } (-\infty, \infty) \] - **Range**: Similarly, the range of a linear function is also all real numbers, as the output \( f(x) \) can take any real value. Therefore, the range is: \[ \text{Range: } (-\infty, \infty) \] ### Summary - **Function**: \( f(x) = 5x + 1 \) - **Domain**: \( (-\infty, \infty) \) - **Range**: \( (-\infty, \infty) \) You can visualize the graph as a straight line with a steep upward slope, crossing the y-axis at (0, 1).