Question

question 1-1
ms. everett asked the class to determine ((hg)(x)) if (h(x) = 4x - 2) and (g(x) = 4x + 2).
two students, marcus and daria, have their work shown below.

marcus	daria
((hg)(x) = 16x^2 - 8x + 8x - 4)	((hg)(x) = 16x + 8 - 2)
((hg)(x) = 16x^2 - 4)	((hg)(x) = 16x + 6)

ms. everett said only one of them has the correct answer.
which student solved the problem incorrectly? explain why the work is incorrect.

Explanation:

Step1: Recall the definition of function composition

The composition \((h \circ g)(x)\) means \(h(g(x))\), which is substituting \(g(x)\) into \(h(x)\) wherever there is an \(x\) in \(h(x)\).

Step2: Analyze Marcus's work

Marcus calculated \((4x + 2)(4x - 2)\), which is \(h(x)\cdot g(x)\) (multiplication of functions) not \(h(g(x))\) (composition of functions). So his approach is incorrect.

Step3: Analyze Daria's work

Given \(h(x)=4x - 2\) and \(g(x)=4x + 2\), to find \(h(g(x))\), we substitute \(g(x)=4x + 2\) into \(h(x)\). So \(h(g(x)) = 4(4x + 2)-2\). Then, distributing the 4: \(16x + 8 - 2\), and combining like terms: \(16x + 6\). Daria followed the correct process for function composition.

Answer:

Marcus solved the problem incorrectly. He computed the product of the functions \(h(x)\) and \(g(x)\) (i.e., \(h(x)\cdot g(x)\)) instead of the composition \(h(g(x))\). The composition \((h \circ g)(x)\) requires substituting \(g(x)\) into \(h(x)\), which Daria correctly performed, while Marcus multiplied the two functions' expressions.