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

To find the value of f(1), we start by using the information that the tangent line at the point (1, 7) also passes through the point (2, 2).

First, we need to find the slope of the tangent line. The slope (m) can be calculated using the formula:

m = (y2 – y1) / (x2 – x1)

Substituting in the points (1, 7) and (2, 2):

m = (2 – 7) / (2 – 1) = -5 / 1 = -5

Now, we have the slope of the tangent line, which is -5. The equation of the tangent line in point-slope form is given by:

y – y1 = m(x – x1)

Substituting (1, 7) and m = -5 into this equation:

y – 7 = -5(x – 1)

Now we can simplify this:

y – 7 = -5x + 5

y = -5x + 12

Next, we can verify the tangent line by checking if it passes through the point (2, 2):

Substituting x = 2 into the line equation:

y = -5(2) + 12 = -10 + 12 = 2

This confirms that the point (2, 2) lies on the tangent line.

Since we derived the equation of the tangent line, we can now conclude that the tangent line touches the function f at the point (1, 7). Therefore, given that the y-coordinate of that point is 7, we can determine:

f(1) = 7