Question

Answer any one question: SECTION B (5 Marks x 1 = 5)

1. Given f(x) = x - 1 and g(x) = x² + 2 * x-8. Find f(g(x)) and g(f(x)). Check whether the statement f(g(x)) = g(f(x)) is true or false.

2. Given f(x) = x² + 7 and g(x) = x - 3. Find f(g(x)) and g(f(x)). Check the whether statement f(g(x)) = g(f(x)) is true or false.​

Answer 1

2. False

Explanation :

given that f(x) = x^2 + 7 and g(x) = x-3

thus, f(g(x)) would be

• f(g(x)) = f(x-3) = (x-3)^2 + 7
• f(g(x)) = (x-3)(x-3) +7
• f(g(x)) = (x(x-3)-3(x-3)) +7
• f(g(x)) = x^2 -3x -3x +9 + 7
• f(g(x)) = x^2 -6x + 16

thus, f(g(x)) = x^2 -6x + 16

Similarly,

g(f(x)) would be

• g(f(x)) = g(x^2 + 7) = x^2 +7 - 3
• g(f(x)) = x^2 + 4

thus, g(f(x)) = x^2 + 4

comparing,

• x^2 -6x + 16 ≠ x^2 + 4

therefore, the statement f(g(x)) = g(f(x)) is false.

Answer 2

`\begin{aligned}\textsf{1.}\quad &f(g(x))=x^2+2x-9\\\\&g(f(x))=x^2-9\\\\&f(g(x))=g(f(x))\; \sf is \;FALSE\end{aligned}`

`\begin{aligned}\textsf{2.}\quad &f(g(x))=x^2-6x+16\\\\&g(f(x))=x^2+4\\\\&f(g(x))=g(f(x))\; \sf is \;FALSE\end{aligned}`

Explanation:

Question 1

Given:

`\begin{cases}f(x)=x-1\\g(x)=x^2+2x-8\end{cases}`

The notation f(g(x)) represents the composition of two functions, f and g. The process involves taking the output of g(x) and substituting it as the input of f(x):

`\begin{aligned}f(g(x))&=f(x^2+2x-8)\\&=(x^2+2x-8)-1\\&=x^2+2x-9\end{aligned}`

The notation g(f(x)) represents the composition of two functions, g and f. The process involves taking the output of f(x) and substituting it as the input of g(x):

`\begin{aligned}g(f(x))&=g(x-1)\\&=(x-1)^2+2(x-1)-8\\&=x^2-2x+1+2x-2-8\\&=x^2-9\end{aligned}`

As x² + 2x - 9 ≠ x² - 9, then the statement f(g(x)) = g(f(x)) is false.

`\hrulefill`

Question 2

Given:

`\begin{cases}f(x)=x^2+7\\g(x)=x-3\end{cases}`

The notation f(g(x)) represents the composition of two functions, f and g. The process involves taking the output of g(x) and substituting it as the input of f(x):

`\begin{aligned}f(g(x))&=f(x-3)\\&=(x-3)^2+7\\&=x^2-6x+9+7\\&=x^2-6x+16\end{aligned}`

The notation g(f(x)) represents the composition of two functions, g and f. The process involves taking the output of f(x) and substituting it as the input of g(x):

`\begin{aligned}g(f(x))&=g(x^2+7)\\&=(x^2+7)-3\\&=x^2+7-3\\&=x^2+4\end{aligned}`

As x² - 6x + 16 ≠ x² + 4, then the statement f(g(x)) = g(f(x)) is false.