Replace f(x) with y.
y = x²-3
Swap the variables.
x = y²-3
Solve for y.
Rewrite the equation as y² - 3x = x.
Add 3 to both sides of the equation.
y² = x - 3
Take the square root on both sides of the equation to eliminate the exponent on the left side.
[tex]\bf{y=\pm\sqrt{x+3} }[/tex]
The complete solution is the result of the positive or negative portions of the solution.
First, use the positive value of ± to find the first solution.
[tex]\bf{y=\sqrt{x+3} }[/tex]
Then use the negative value of ± to find the second solution.
[tex]\bf{y=-\sqrt{x+3} }[/tex]
The complete solution is the result of the positive or negative portions of the solution.
[tex]\bf{y=\sqrt{x+3} }[/tex]
[tex]\bf{y=-\sqrt{x+3} }[/tex]
Solve for y and substitute with [tex]\bf{f^{-1}(x) }[/tex].
substitute f⁻¹( x ) for y to show the final answer.
[tex]\boldsymbol{f^{-1}(x)=\sqrt{x+3},-\sqrt{x+3} }[/tex]
Set the composite results function.
[tex]\boldsymbol{g(f(x)) }[/tex]
Evaluate g(f(x)) by substituting the value of f into g.
[tex]\boldsymbol{\sqrt{(x^{2} -3)+3 } }[/tex]
Add −3 and 3.
[tex]\boldsymbol{g(x^{2} -3)=\sqrt{x^{2} +0} }[/tex]
Add x² and 0.
[tex]\boldsymbol{g(x^{2} -3)=\sqrt{x^{2}} }[/tex]
Extract terms from under the radical, assuming positive real numbers.
[tex]\boldsymbol{g(x^{2} -3)=x }[/tex]
Given the [tex]\bf{g(f(x))=x,f^{-1}(x)=\sqrt{x+3},-\sqrt{x+3} }[/tex] is the inverse of [tex]\bf{f(x)=x^{2} -3. }[/tex]
[tex]\boldsymbol{f^{-1}(x)=\sqrt{x+3},-\sqrt{x+3}. }[/tex]
