The function $f(x) = 7x + 4$ is a linear function. Here's some analysis:

Key Characteristics:

1. Slope (Rate of Change): The slope is $7$. For every 1 unit increase in $x$, $f(x)$ increases by $7$.
2. Y-intercept: The $y$-intercept is $4$. This is the value of $f(x)$ when $x = 0$, so the graph of $f(x)$ passes through the point $(0, 4)$.
3. Domain: The domain is all real numbers ($x \in \mathbb{R}$) because the function is defined for all $x$.
4. Range: The range is also all real numbers ($f(x) \in \mathbb{R}$) because the output covers all possible real values.
5. Graph: The graph is a straight line with a positive slope, increasing as $x$ increases.

Example:

If $x = 2$:
$f(2) = 7(2) + 4 = 14 + 4 = 18$

If $x = -3$:
$f(-3) = 7(-3) + 4 = -21 + 4 = -17$

Let me know if you'd like a graph of this function or further assistance!

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Relative Questions:

1. What is the slope of the function $f(x) = 7x + 4$ and how does it affect the graph?
2. How do you find the $y$-intercept of a linear function, and what does it represent in this case?
3. Can you calculate $f(x)$ for $x = 5$ and $x = -1$?
4. What is the general formula for a linear function and how does $f(x)$ fit into this formula?
5. How would the graph of $f(x) = 7x + 4$ change if the slope were reduced to $3$?

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Tip:

To sketch the graph of a linear function quickly, identify two points (e.g., using $x = 0$ and $x = 1$), plot them, and draw a straight line through!