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Question 515869: f(x) = 10x^2 + 3 find domain and range
Answer by drumwrrv(7)   (Show Source):
You can put this solution on YOUR website! domain
f(x)=10x^(2)+3
The domain of the rational expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
All real numbers
range
f(x)=10x^(2)+3
Since x is on the right-hand side of the equation, switch the sides so it is on the left-hand side of the equation.
10x^(2)+3=f(x)
Since 3 does not contain the variable to solve for, move it to the right-hand side of the equation by subtracting 3 from both sides.
10x^(2)=-3+f(x)
Move all terms not containing x to the right-hand side of the equation.
10x^(2)=f(x)-3
Divide each term in the equation by 10.
(10x^(2))/(10)=(f(x))/(10)-(3)/(10)
Simplify the left-hand side of the equation by canceling the common terms.
x^(2)=(f(x))/(10)-(3)/(10)
Combine the numerators of all expressions that have common denominators.
x^(2)=(f(x)-3)/(10)
Take the square root of both sides of the equation to eliminate the exponent on the left-hand side.
x=\~((f(x)-3)/(10))
Split the fraction inside the radical into a separate radical expression in the numerator and the denominator. A fraction of roots is equivalent to a root of the fraction.
x=\(~(f(x)-3))/(~(10))
To rationalize the denominator of a fraction, rewrite the fraction so that the new fraction has the same value as the original and has a rational denominator. The factor to multiply by should be an expression that will eliminate the radical in the denominator. In this case, the expression will eliminate the radical in the denominator is (~(10))/(~(10)).
x=\(~(f(x)-3))/(~(10))*(~(10))/(~(10))
Simplify the rationalized fraction.
x=\(~(10(f(x)-3)))/(10)
First, substitute in the + portion of the \ to find the first solution.
x=(~(10(f(x)-3)))/(10)
Next, substitute in the - portion of the \ to find the second solution.
x=-(~(10(f(x)-3)))/(10)
The complete solution is the result of both the + and - portions of the solution.
x=(~(10(f(x)-3)))/(10),-(~(10(f(x)-3)))/(10)
The domain of an expression is all real numbers except for the regions where the expression is undefined. This can occur where the denominator equals 0, a square root is less than 0, or a logarithm is less than or equal to 0. All of these are undefined and therefore are not part of the domain.
(10(f(x)-3))<0
Solve the equation to find where the original expression is undefined.
f(x)<3
The domain of the rational expression is all real numbers except where the expression is undefined.
f(x)>=3_[3,I)
The domain of the inverse of f(x)=10x^(2)+3 is equal to the range of f(f(x))=(~(10(f(x)-3)))/(10).
Range: f(x)>=3_[3,infinity)
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