Final answer:
To determine f(4), we first find the slope of the tangent line to the graph of fg at x=4. Then, we use the equation of the tangent line to find the value of y when x=4, which represents the value of f(4).
Step-by-step explanation:
To find f(4), we need to determine the value of y when x=4 on the line tangent to the graph of fg at x=4. The equation of the tangent line can be written in the slope-intercept form as y = mx + b, where m is the slope of the line and b is the y-intercept.
Given that the equation of the tangent line is 20x - 2y = 3, we can rearrange it to the slope-intercept form: y = 10x - (3/2).
Since the equation of the tangent line is also the equation of the line tangent to g at x=4, its slope (m) is equal to the derivative of g at x=4. Therefore, the derivative of g at x=4 is 10. And since the derivative of g represents the rate of change of g, we can conclude that the rate of change of g at x=4 is 10. This means that for every unit increase in x, the corresponding increase in g is 10 units.
Since x=4 is an x-intercept of g, this means that g(4) = 0. And since the rate of change of g at x=4 is 10, this means that when x increases by 1 unit, g increases by 10 units. Therefore, when x=4, g is equal to 0, and when x=5, g is equal to 10. So, the value of g increases by 10 units when x increases by 1 unit.
Now, let's turn our attention to the function fg. The equation of the tangent line to the graph of fg at x=4 is also the equation of the line tangent to g at x=4. Therefore, the slope of the tangent line of fg at x=4 is 10. This means that for every unit increase in x, the corresponding increase in fg is 10 units.
Since the equation of the tangent line to fg at x=4 is given as 20x - 2y = 3, we can substitute x=4 and solve for y to find the value of y when x=4: 20(4) - 2y = 3, which simplifies to 80 - 2y = 3. Solving for y, we get y = 38.5.
Therefore, when x=4, the value of y on the graph of fg is 38.5. As mentioned earlier, the value of y corresponds to the value of f(x). So, f(4) = 38.5.