Question

\(y(x) = mx + c\) represents a linear relation in y and x. If \(y(-1) = 3\) and \(y(3) = 11\), find the value of \(y(y(2))\)

Hint:

For composite function evaluation like \(f(g(x))\), always work from the inside out. First calculate the value of the inner function \(g(x)\), and then use that result as the input for the outer function \(f\).

Answer 1

Step 1: Understanding the Concept:
We are given a linear function and two points that lie on the line. We need to first determine the equation of the line (i.e., find the values of \(m\) and \(c\)) and then evaluate a composite function.
Step 2: Detailed Explanation:
The given function is \(y(x) = mx + c\).
We are given two points:

When \(x = -1\), \(y = 3\). So, \(y(-1) = 3\).
When \(x = 3\), \(y = 11\). So, \(y(3) = 11\).
Find the slope (m): The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). \[ m = \frac{11 - 3}{3 - (-1)} = \frac{8}{4} = 2 \] Find the y-intercept (c): Now we know the equation is \(y(x) = 2x + c\). We can use one of the given points to find \(c\). Let's use \((-1, 3)\). \[ 3 = 2(-1) + c \] \[ 3 = -2 + c \] \[ c = 5 \] So, the linear function is \(y(x) = 2x + 5\).
Evaluate \(y(y(2))\): This is a composite function evaluation. We first evaluate the inner function, \(y(2)\). \[ y(2) = 2(2) + 5 = 4 + 5 = 9 \] Now we substitute this result into the outer function. We need to find \(y(9)\). \[ y(9) = 2(9) + 5 = 18 + 5 = 23 \] Step 3: Final Answer:
The value of \(y(y(2))\) is 23.