The function f is defined by f  x   = 3x + k, where k is a constant. Find the value of k if the graph of f passes through the point (−1, 8).

1. Break Down the Problem

We need to find the value of the constant $k$ in the function $f(x) = 3x + k$, given that the graph passes through the point $(-1, 8)$. This means that when $x = -1$, $f(x) = 8$.

2. Relevant Concepts

To solve for $k$, we will substitute the coordinates of the point into the function and set it equal to 8: $f(-1) = 3(-1) + k = 8$

3. Analysis and Detail

Now, we will calculate:

$f(-1) = 3(-1) + k$ $= -3 + k$ Setting this equal to 8, we have: $-3 + k = 8$

Now, solve for $k$: $k = 8 + 3$ $k = 11$

4. Verify and Summarize

We can verify by plugging $k = 11$ back into the function and checking if the point (-1, 8) is on the graph. $f(-1) = 3(-1) + 11 = -3 + 11 = 8$ This confirms that our calculations and logic are correct.

Final Answer

The value of $k$ is $11$.