If the line tangent to the graph of the function f at the point (1, 7) also passes through the point (2, 2), what is the value of f(1)?

To solve this problem, we first need to understand the meaning of a tangent line. A tangent line to the graph of a function at a specific point is a straight line that just ‘touches’ the graph at that point and has the same slope as the function at that point.

In this case, we know the tangent line touches the function f at the point (1, 7) and also passes through the point (2, 2). This gives us two points: A(1, 7) and B(2, 2).

Next, we can determine the slope of the line passing through these two points using the slope formula:

Slope (m) = (y₂ – y₁) / (x₂ – x₁)

Substituting the coordinates of points A and B into the formula:

m = (2 – 7) / (2 – 1) = -5 / 1 = -5

This slope of -5 represents the slope of the tangent line at the point (1, 7). Thus, we can say:

f'(1) = -5

Now we know the slope at x = 1, but we also need to confirm the value of f(1). Since we have the point (1, 7), it tells us directly that:

f(1) = 7

Hence, the value of f(1) is 7.

In conclusion, the answer to the question is:

f(1) = 7