Notes
Composite Functions
\((f \circ g)(x) = f(g(x))\)
Practice Problems
\(\textbf{1)}\) \(f(x)=x^2\) and \( g(x)=x+4. \, \)
What is \( (f\circ g)(x)\)?
\(\text{The answer is }(f \circ g)(x)=x^2+8x+16\)
\(\,\,\,\,\,\,(f\circ g)(x)\)
\(\,\,\,\,\,\,(f(g(x))\)
\(\,\,\,\,\,\,f(x+4)\)
\(\,\,\,\,\,\,(x+4)^2\)
\(\,\,\,\,\,\,x^2+8x+16\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ g)(x)=x^2+8x+16\)
![Link to Youtube Video Solving Question Number 1](https://andymath.com/wp-content/uploads/2020/08/play-video.jpg")
\(\textbf{2)}\) \(f(x)=x^2\) and \( g(x)=x+4. \, \)
What is \( (g\circ f)(x)\)?
The answer is \((g \circ f)(x)=x^2+4\)
\(\,\,\,\,\,\,(g\circ f)(x)\)
\(\,\,\,\,\,\,g(f(x))\)
\(\,\,\,\,\,\,g(x^2)\)
\(\,\,\,\,\,\,(x^2)+4\)
\(\,\,\,\,\,\,x^2+4\)
\(\,\,\,\,\,\,\)The answer is \((g \circ f)(x)=x^2+4\)
\(\textbf{3)}\) \(f(x)=x^2\) and \( g(x)=x+4. \, \)
What is \( (f\circ f)(x)\)?
The answer is \((f \circ f)(x)=x^4\)
\(\,\,\,\,\,\,(f\circ f)(x)\)
\(\,\,\,\,\,\,f(f(x))\)
\(\,\,\,\,\,\,f(x^2)\)
\(\,\,\,\,\,\,(x^2)^2\)
\(\,\,\,\,\,\,x^4\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ f)(x)=x^4\)
\(\textbf{4)}\) \(f(x)=x^2\) and \( g(x)=x+4. \, \)
What is \( (g\circ g)(x)\)?
The answer is \((g \circ g)(x)=x+8\)
\(\,\,\,\,\,\,(g\circ g)(x)\)
\(\,\,\,\,\,\,g(g(x))\)
\(\,\,\,\,\,\,g(x+4)\)
\(\,\,\,\,\,\,(x+4)+4\)
\(\,\,\,\,\,\,x+8\)
\(\,\,\,\,\,\,\text{The answer is }(g \circ g)(x)=x+8\)
\(\textbf{5)}\) \(f(x)=3x-4 \) and \( g(x)=1-x^2 \)
What is \(f(g(2))\)?
The answer is \(-13 \)
\(\,\,\,\,\,\,f(g(2))\)
\(\,\,\,\,\,\,f(1-(2)^2)\)
\(\,\,\,\,\,\,f(1-4)\)
\(\,\,\,\,\,\,f(-3)\)
\(\,\,\,\,\,\,3(-3)-4\)
\(\,\,\,\,\,\,-9-4\)
\(\,\,\,\,\,\,-13\)
\(\,\,\,\,\,\,\text{The answer is }-13\)
\(\textbf{6)}\) \(f(x)=2x-4 \) and \( g(x)=x^2-1 \)
What is \(\left(f \circ g\right) (3)\)?
The answer is \(12 \)
\(\,\,\,\,\,\,(f \circ g)(3)\)
\(\,\,\,\,\,\,f(g(3))\)
\(\,\,\,\,\,\,f((3)^2-1)\)
\(\,\,\,\,\,\,f(9-1)\)
\(\,\,\,\,\,\,f(8)\)
\(\,\,\,\,\,\,2(8)-4\)
\(\,\,\,\,\,\,16-4\)
\(\,\,\,\,\,\,12\)
\(\,\,\,\,\,\,\text{The answer is }12\)
\(\textbf{7)}\) \(f(x)=3x^2\) and \( g(x)=x+1. \, \)
What is \( (f\circ g)(2)\)?
The answer is \((f \circ g)(2)=27\)
\(\,\,\,\,\,\,(f\circ g)(2)\)
\(\,\,\,\,\,\,f(g(2))\)
\(\,\,\,\,\,\,f(2+1)\)
\(\,\,\,\,\,\,f(3)\)
\(\,\,\,\,\,\,3(3)^2\)
\(\,\,\,\,\,\,3(9)\)
\(\,\,\,\,\,\,27\)
\(\,\,\,\,\,\,\text{The answer is }27\)
\(\textbf{8)}\) \(f(x)=x^2-1\) and \( g(x)=3x-9. \, \)
What is \( (f\circ g)(0)\)?
The answer is \((f \circ g)(0)=80\)
\(\,\,\,\,\,\,(f\circ g)(0)\)
\(\,\,\,\,\,\,f(g(0))\)
\(\,\,\,\,\,\,f(3(0)-9)\)
\(\,\,\,\,\,\,f(-9)\)
\(\,\,\,\,\,\,(-9)^2-1\)
\(\,\,\,\,\,\,81-1\)
\(\,\,\,\,\,\,80\)
\(\,\,\,\,\,\,\text{The answer is }80\)
\(\textbf{9)}\) \(f(x)=4x+3\) and \( g(x)=x^2+4. \, \)
What is \( (f\circ g)(x)\)?
The answer is \((f \circ g)(x)=4x^2+19\)
\(\,\,\,\,\,\,(f\circ g)(x)\)
\(\,\,\,\,\,\,f(g(x))\)
\(\,\,\,\,\,\,f(x^2+4)\)
\(\,\,\,\,\,\,4(x^2+4)+3\)
\(\,\,\,\,\,\,4x^2+16+3\)
\(\,\,\,\,\,\,4x^2+19\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ g)(x)=4x^2+19\)
\(\textbf{10)}\) \(f(x)=8x-1\) and \( g(x)=x+2. \, \)
What is \( (f\circ f)(x)\)?
The answer is \((f \circ f)(x)=64x-9\)
\(\,\,\,\,\,\,(f\circ f)(x)\)
\(\,\,\,\,\,\,f(f(x))\)
\(\,\,\,\,\,\,f(8x-1)\)
\(\,\,\,\,\,\,8(8x-1)-1\)
\(\,\,\,\,\,\,64x-8-1\)
\(\,\,\,\,\,\,64x-9\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ f)(x)=64x-9\)
\(\textbf{11)}\) \(f(x)=2x^2\) and \( g(x)=3x+1. \, \)
What is \( (g\circ f)(x)\)?
The answer is \((g \circ f)(x)=6x^2+1\)
\(\,\,\,\,\,\,(g\circ f)(x)\)
\(\,\,\,\,\,\,g(f(x))\)
\(\,\,\,\,\,\,g(2x^2)\)
\(\,\,\,\,\,\,3(2x^2)+1\)
\(\,\,\,\,\,\,6x^2+1\)
\(\,\,\,\,\,\,\text{The answer is }(g \circ f)(x)=6x^2+1\)
\(\textbf{12)}\) \(f(x)=x^3\) and \( g(x)=2x-4. \, \)
What is \( (f\circ f)(x)\)?
The answer is \((f \circ f)(x)=x^9\)
\(\,\,\,\,\,\,(f\circ f)(x)\)
\(\,\,\,\,\,\,f(f(x))\)
\(\,\,\,\,\,\,f(x^3)\)
\(\,\,\,\,\,\,(x^3)^3\)
\(\,\,\,\,\,\,x^9\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ f)(x)=x^9\)
\(\textbf{13)}\) \(f(x)=x^2\) and \( g(x)=x+1. \, \)
What is \( (f\circ g)(x)\)?
The answer is \((f \circ g)(x)=x^2+2x+1\)
\(\,\,\,\,\,\,(f\circ g)(x)\)
\(\,\,\,\,\,\,f(g(x))\)
\(\,\,\,\,\,\,f(x+1)\)
\(\,\,\,\,\,\,(x+1)^2\)
\(\,\,\,\,\,\,x^2+2x+1\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ g)(x)=x^2+2x+1\)
\(\textbf{14)}\) \(f(x)=x-1\) and \( g(x)=x-8. \, \)
What is \( (f\circ g)(3)\)?
The answer is \((f \circ g)(3)=-6\)
\(\,\,\,\,\,\,(f\circ g)(3)\)
\(\,\,\,\,\,\,f(g(3))\)
\(\,\,\,\,\,\,f(3-8)\)
\(\,\,\,\,\,\,(3-8)-1\)
\(\,\,\,\,\,\,-6\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ g)(3)=-6\)
\(\textbf{15)}\) \(f(x)=5x^2-1\) and \( g(x)=x. \, \)
What is \( (f\circ g)(x)\)?
The answer is \((f \circ g)(x)=5x^2-1\)
\(\,\,\,\,\,\,(f\circ g)(x)\)
\(\,\,\,\,\,\,f(g(x))\)
\(\,\,\,\,\,\,f(x)\)
\(\,\,\,\,\,\,5x^2-1\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ g)(x)=5x^2-1\)
\(\textbf{16)}\) \(f(x)=3x^2\) and \( g(x)=x+2. \, \)
What is \( (f\circ f)(x)\)?
The answer is \((f \circ f)(x)=27x^4\)
\(\,\,\,\,\,\,(f\circ f)(x)\)
\(\,\,\,\,\,\,f(f(x))\)
\(\,\,\,\,\,\,f(3x^2)\)
\(\,\,\,\,\,\,3(3x^2)^2\)
\(\,\,\,\,\,\,3(9x^4)\)
\(\,\,\,\,\,\,27x^4\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ f)(x)=27x^4\)
\(\textbf{17)}\) \(f(x)=3x^2\) and \( g(x)=x+1. \, \)
What is \( (f\circ g)(x)\)?
The answer is \((f \circ g)(x)=3x^2+6x+3\)
\(\,\,\,\,\,\,(f\circ g)(x)\)
\(\,\,\,\,\,\,f(g(x))\)
\(\,\,\,\,\,\,f(x+1)\)
\(\,\,\,\,\,\,3(x+1)^2\)
\(\,\,\,\,\,\,3(x^2+2x+1)\)
\(\,\,\,\,\,\,3x^2+6x+3\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ g)(x)=3x^2+6x+3\)
\(\textbf{18)}\) \(f(x)=x^2-1\) and \( g(x)=3x-9. \, \)
What is \( (f\circ g)(x)\)?
The answer is \((f \circ g)(x)=9x^2-54x+80\)
\(\,\,\,\,\,\,(f\circ g)(x)\)
\(\,\,\,\,\,\,f(g(x))\)
\(\,\,\,\,\,\,f(3x-9)\)
\(\,\,\,\,\,\,(3x-9)^2-1\)
\(\,\,\,\,\,\,(9x^2-54x+80)\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ g)(x)=9x^2-54x+80\)
\(\textbf{19)}\) \(f(x)=4x+3\) and \( g(x)=x^2+4. \, \)
What is \( (f\circ g)(1)\)?
The answer is \((f \circ g)(1)=23\)
\(\,\,\,\,\,\,(f\circ g)(1)\)
\(\,\,\,\,\,\,f(g(1))\)
\(\,\,\,\,\,\,f(1^2+4)\)
\(\,\,\,\,\,\,4(1^2+4)+3\)
\(\,\,\,\,\,\,4(5)+3\)
\(\,\,\,\,\,\,20+3\)
\(\,\,\,\,\,\,23\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ g)(1)=23\)
\(\textbf{20)}\) \(f(x)=8x-1\) and \( g(x)=x+2. \, \)
What is \( (f\circ f)(5)\)?
The answer is \((f \circ f)(5)=311\)
\(\,\,\,\,\,\,(f\circ f)(5)\)
\(\,\,\,\,\,\,f(f(5))\)
\(\,\,\,\,\,\,f(8(5)-1)\)
\(\,\,\,\,\,\,f(39)\)
\(\,\,\,\,\,\,8(39)-1\)
\(\,\,\,\,\,\,311\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ f)(5)=311\)
\(\textbf{21)}\) \(f(x)=2x^2\) and \( g(x)=3x+1. \, \)
What is \( (g\circ f)(4)\)?
The answer is \((g \circ f)(4)=97\)
\(\,\,\,\,\,\,(g\circ f)(4)\)
\(\,\,\,\,\,\,g(f(4))\)
\(\,\,\,\,\,\,g(2(4)^2)\)
\(\,\,\,\,\,\,3(2(4)^2)+1\)
\(\,\,\,\,\,\,3(32)+1
[latex]\,\,\,\,\,\,97\)
\(\,\,\,\,\,\,\text{The answer is }(g \circ f)(4)=97\)
\(\textbf{22)}\) \(f(x)=x^3\) and \( g(x)=2x-4. \, \)
What is \( (f\circ f)(2)\)?
The answer is \((f \circ f)(2)=512\)
\(\,\,\,\,\,\,(f\circ f)(2)\)
\(\,\,\,\,\,\,f(f(2))\)
\(\,\,\,\,\,\,f(2^3)\)
\(\,\,\,\,\,\,f(8)\)
\(\,\,\,\,\,\,8^3\)
\(\,\,\,\,\,\,512\)
\(\,\,\,\,\,\,\text{The answer is }(f \circ f)(2)=512\)
\(\textbf{23)}\) \(f(x)=x^2\) and \( g(x)=x+1. \, \)
What is \( (f\circ g)(3)\)?
The answer is \((f \circ g)(3)=16\)
\(\textbf{24)}\) \(f(x)=x-1\) and \( g(x)=x-8. \, \)
What is \( (f\circ g)(x)\)?
The answer is \((f \circ g)(x)=x-9\)
\(\textbf{25)}\) \(f(x)=5x^2-1\) and \( g(x)=x. \, \)
What is \( (f\circ g)(-2)\)?
The answer is \((f \circ g)(-2)=19\)
\(\textbf{26)}\) \(f(x)=3x^2\) and \( g(x)=x+2. \, \)
What is \( (f\circ f)(1)\)?
The answer is \((f \circ f)(1)=27\)
True or False?
\(\textbf{27)}\) \((f\circ g)(x)=(g\circ f)(x)\)
The statement is false.
Challenge Problems
For questions 28-31, use the following functions.
\( f(x)=8-x \) Domain:\(\, 3\le x \le 8 \)
\( g(x)=x+3 \) Domain:\(\, 1\le x \le 6 \)
\(\textbf{28)}\) \( (f\circ g)(3) \)
The answer is \( 2 \)
\(\textbf{29)}\) \( (f\circ g)(6) \)
The answer is undefined
\(\textbf{30)}\) \( (f\circ g)(x) \)
The answer is \( 5-x \)
\(\textbf{31)}\) Find the domain for \( (f\circ g)(x) \)
The answer is \( 1\le x\le 5 \)
\(\textbf{32)}\) \(f(x)=4x-4 \) and \( g(x)=1-x^2, \)
What is \(g(f(n+1))\)?
The answer is \(1-16n^2 \)
\(\textbf{33)}\) \(f(x)=4x-1\) and \( g(x)=x^2 \)
Solve for the variable \(m. f(g(m))=15 \)
The answer is \(m=\pm 2 \)
\(\textbf{34)}\) \((f\circ g)(x)=x^2+3\) and \( f(x)=x+4. \, \)
What is \( g(x)\)?
The answer is \(g(x)=x^2-1\)
\(\textbf{35)}\) \(f(x)=8x+3\) and \( g(x)=x+2\)
What is \( \displaystyle f\left(\frac{1}{g(2)}\right)\)?
The answer is \(\displaystyle f\left(\frac{1}{g(2)}\right)=5\)
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