Question

Ted argues that the solution to the equation f(x) = g(x) is the x-value of their intersection point, while Lisa argues that the solution is the y-value of their intersection point. They decide to test their conjecture by finding the value of f(x) = g(x) when f(x) = 3x−5 and g(x) = x³−3x²+4x . Enter the value of f(x) = g(x) in the box below.

Answer 1

The value of f(x) = g(x) is -8, which occurs when x = 5. This means that the intersection point of the two functions is at (5, -8).

To find the solution to the equation f(x) = g(x), we need to solve for x. This means we need to find the value of x where both functions have the same output (y-value).

Here's how we can solve:

Step 1: Set the equations equal to each other:

f(x) = g(x)

3x - 5 = x³ - 3x² + 4x

Step 2: Combine like terms:

-3x² - x + 5 = 0

Step 3: Factor the equation (by grouping or using the quadratic formula):

(3x - 1)(x - 5) = 0

Step 4: Solve for x:

3x - 1 = 0 or x - 5 = 0

x = 1/3 or x = 5

Step 5: Evaluate f(x) or g(x) at the solutions to see if they are equal.

When x = 1/3:

f(1/3) = 3(1/3) - 5 = -4

When x = 5:

g(5) = 5^3 - 3(5^2) + 4(5) = -8

So, while Ted is correct that the solution to the equation is the x-value of the intersection point, Lisa is incorrect as the solution is not the y-value but the x-value where both functions have the same output.