Determine whether the function f is a bijection from R to R ? Find fog and gof where f(x)=2x2+3 and g(x)=x=1
Given functions, "f(x)=2x^2+3,g(x)=x-1"
Let A and B be two sets of real numbers.
Let "x_1,x_2 \\in A" such that "f(x_1)=f(x_2)"
"\\Rightarrow 2x_1^2+3=2x_2^2+3\n\n\\\\\n\n\\Rightarrow x_1^2=x_2^2\n\n\\\\\n\n\\Rightarrow x_1^2-x_2^2=0\n\n\\\\\n\n\\Rightarrow (x_1-x_2)(x_1+x_2)=0\n\n\\\\\n\n\\Rightarrow x_1=\\pm x_2. Thus f(x_1)=f(x_2) \\text{ does not implies that }x_1=x" _2.
For instance, f(1)=f(-1)=2 i.e. Two element have the same image. So F is many one function.
Now , let "y=2x^2+3\\Rightarrow x=\\sqrt{\\dfrac{y-3}{2}}"
Elements of y have no pre-image in A ( for instance an element -2 in the codomain has no pre image in domain A. SO f is not onto.
So F is not bijective.
"fog(x)=f(x-1)=2(x-1)^2+3=2x^2+2-4x+3=2x^2-4x+5"
"gof(x)=g(2x^2+3)=2x^2+3-1=2^2+2"
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