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Functions, 2ADV 2025 MET2 13 MC

The graphs of  \(y=f(x)\)  and  \(y=g(x)\)  are sketched on the same set of axes below.
 

  

Which of the following could be the graph of  \(y=g(f(x))\) ?
 

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\(C\)

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\(\text{Consider possible functions for each graph.}\)

\(y=f(x) \ \Rightarrow \ y=-x\)

\(y=g(x) \ \Rightarrow \ y=x(x+1)(x-1)(x-2)\)

\(y=g(f(x))\) \(=-x(-x+1)(-x-1)(-x-2) \)  
  \(=x(1-x)(x+1)(-x-2) \)  
  \(=-x(1-x)(x+1)(x+2) \)  

 
\(\text{By elimination:}\)

\(\text{Graph cuts \(x\)-axis at}\ \ x=0, 1, -1, -2\ \ \text{(eliminate A and D)}\)

\(\text{At}\ \ x=\dfrac{1}{2},\ g(f(x))<0\ \ \text{(eliminate B)}\)

\(\Rightarrow C\)

♦ Mean mark 45%.

Functions, 2ADV EQ-Bank 16

  1. Given the functions  \(f(x)=x-2\)  and  \(g(x)=x^2\), determine the equation of \(f(g(x))\) and sketch its graph showing all intercepts.   (2 marks) 

  2. Evaluate \(g(f(-1))\).   (1 mark)

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a.   

b.  \(g(f(-1))=9\)

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a.    \(f(x)=x-2, \ g(x)=x^2\)

\(f(g(x))=x^2-2\)

\(\text{Intercepts:} \ \ x^2-2=0 \ \Rightarrow \ x= \pm \sqrt{2}\)
 

b.    \(f(-1)=-1-2=-3\)

\(g(f(-1))=g(-3)=9\)

Functions, 2ADV F1 2022 HSC 10 MC

The graphs of `y=f(x)` and `y=g(x)` are shown.
 

Which graph best represents  `y=g(f(x))`
 

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`B`

Show Worked Solution

`text{By Elimination:}`

`y=f(x)\ \ text{is an even function}`

`y=f(x)=f(-x)\ \ =>\ \ g(f(x))=g(f(-x))`

`g(f(x))\ \ text{is also an even function}`

`text{→ Eliminate A, C and D}`

`=>B`


♦ Mean mark 46%.

Functions, 2ADV F1 EQ-Bank 21

Given the function  `f(x) = sqrt(3-x)`  and  `g(x) = x^2-2`, sketch  `y = g(f(x))`  over its natural domain.   (2 marks)

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Show Worked Solution

`g(x) = x^2-2,\ \ f(x) = sqrt(3-x)`

`g(f(x))` `= (sqrt(3-x))^2-2`
  `= 3-x-2`
  `= 1-x`

 
`text(S)text(ince)\ \ f(x) = sqrt(3-x),`

`=> text(Domain:)\ x <= 3`