Find the inverse g of the following functions:
For x≥0, f\left(x\right)=3x^{2}-4
In order to find the inverse of a function f, set f\left(x\right)=y and solve the equation for x. The unique solution of the equation x=g\left(y\right) is the expression of the inverse of the function f.
In the problem:
f\left(x\right)=y
3x^{2}-4=y
Solve this equation for x :
3x^{2}=y+4
x^{2}=\dfrac{y+4}{3}
The equation has a solution if and only if:
\dfrac{y+4}{3}\geqslant0
Which is equivalent to:
y\geqslant-4
If y\geqslant-4, then the solution of the equation is:
x=\sqrt{\dfrac{y+4}{3}} or x=-\sqrt{\dfrac{y+4}{3}}
Since x\geqslant0, we can conclude that:
x=\sqrt{\dfrac{y+4}{3}}
We can conclude that the inverse of the function f is:
g\left(y\right)=\sqrt{\dfrac{y+4}{3}}, where y\geqslant-4
Replacing y with x leads to:
g\left(x\right)=\sqrt{\dfrac{x+4}{3}}, where x\geqslant-4
g is defined for x≥-4, and g\left(x\right)=\sqrt{\dfrac{x+4}{3}}.
For x≥3, f\left(x\right)=2\sqrt{x-3}+7
In order to find the inverse of a function f, set f\left(x\right)=y and solve the equation for x. The unique solution of the equation x=g\left(y\right) is the expression of the inverse of the function f.
In our problem:
f\left(x\right)=y
2\sqrt{x-3}+7=y
Solve this equation for x :
2\sqrt{x-3}=y-7
\sqrt{x-3}=\dfrac{y-7}{2}
The equation has a solution if and only if:
\dfrac{y-7}{2}\geqslant0
Which is equivalent to:
y\geqslant7
If y\geqslant7, then the solution of the equation is:
x-3=\left( \dfrac{y-7}{2} \right)^{2}
which is equivalent to:
x=\left( \dfrac{y-7}{2} \right)^{2}+3
We can conclude that the inverse of the function f is:
g\left(y\right)=\left( \dfrac{y-7}{2} \right)^{2}+3 where y\geqslant7
Replacing y with x leads to:
g\left(x\right)=\left( \dfrac{x-7}{2} \right)^{2}+3 where x\geqslant7
g is defined for x≥7, and g\left(x\right)=\left( \dfrac{x-7}{2} \right)^{2}+3.
For x \gt 8, f\left(x\right)=3\log_5\left(x-8\right)+1
In order to find the inverse of a function f, set f\left(x\right)=y and solve the equation for x. The unique solution of the equation x=g\left(y\right) is the expression of the inverse of the function f.
In our problem:
f\left(x\right)=y
3\log_5\left(x-8\right)+1=y
Solve this equation for x :
3\log_5\left(x-8\right)=y-1
\log_5\left(x-8\right)=\dfrac{y-1}{3}
x-8=5^{\frac{y-1}{3}}
x=5^{\frac{y-1}{3}}+8
We can conclude that the inverse of the function f is:
g\left(y\right)=5^{\frac{y-1}{3}}+8 where y\in\mathbb{R}
Replacing y with x leads to:
g\left(x\right)=5^{\frac{x-1}{3}}+8 where x\in\mathbb{R}.
g is defined for x\in\mathbb{R}, and g\left(x\right)=5^{\frac{x-1}{3}}+8.
For x\in \mathbb{R}, f\left(x\right)=4e^{3x}-9
In order to find the inverse of a function f, set f\left(x\right)=y and solve the equation for x. The unique solution of the equation x=g\left(y\right) is the expression of the inverse of the function f.
In our problem:
f\left(x\right)=y
4e^{3x}-9=y
Solve this equation for x :
4e^{3x}=y+9
e^{3x}=\dfrac{y+9}{4}
The equation has a solution if and only if:
\dfrac{y+9}{4}\geqslant0
Which is equivalent to:
y\geqslant-9
If y\geqslant-9, then the solution of the equation is:
3x=\ln\left(\dfrac{y+9}{4}\right)
x=\dfrac{1}{3}\ln\left(\dfrac{y+9}{4}\right)
We can conclude that the inverse of the function f is:
g\left(y\right)=\dfrac{1}{3}\ln\left(\dfrac{y+9}{4}\right), where y\geqslant-9
Replacing y with x leads to:
g\left(x\right)=\dfrac{1}{3}\ln\left(\dfrac{x+9}{4}\right), where x\geqslant-9.
g is defined for x\geqslant-9, and g\left(x\right)=\dfrac{1}{3}\ln\left(\dfrac{x+9}{4}\right).
For x\neq-3, f\left(x\right)=\dfrac{2x-4}{x+3}
In order to find the inverse of a function f, set f\left(x\right)=y and solve the equation for x. The unique solution of the equation x=g\left(y\right) is the expression of the inverse of the function f.
In our problem:
f\left(x\right)=y
\dfrac{2x-4}{x+3}=y
Solve this equation for x :
2x-4=\left(x+3\right)y
2x-4=xy+3y
2x-xy=3y+4
x\left(2-y\right)=3y+4
The equation has a solution if and only if:
2-y\neq0
Which is equivalent to:
y\neq2
If y\neq2, then the solution of the equation is:
x=\dfrac{3y+4}{2-y}
We can conclude that the inverse of the function f is:
g\left(y\right)=\dfrac{3y+4}{2-y} where y\neq2
Replacing y with x leads to:
g\left(x\right)=\dfrac{3x+4}{2-x} where x\neq2.
g is defined for x\neq2, and g\left(x\right)=\dfrac{3x+4}{2-x}.
For x\in\mathbb{R}, f\left(x\right)=5x^{3}+9
In order to find the inverse of a function f, set f\left(x\right)=y and solve the equation for x. The unique solution of the equation x=g\left(y\right) is the expression of the inverse of the function f.
In our problem:
f\left(x\right)=y
5x^{3}+9=y
Solve this equation for x :
5x^{3}=y-9
x^{3}=\dfrac{y-9}{5}
x=\sqrt[3]{\frac{y-9}{5}}
We can conclude that the inverse of the function f is:
g\left(y\right)=\sqrt[3]{\frac{y-9}{5}} where y\in\mathbb{R}
Replacing y with x leads to:
g\left(x\right)=\sqrt[3]{\frac{x-9}{5}} where x\in\mathbb{R}.
g is defined for x\in\mathbb{R}, and g\left(x\right)=\sqrt[3]{\frac{x-9}{5}}.
For x\in\mathbb{R}, f\left(x\right)=4\sqrt[5]{2x+3}-7
In order to find the inverse of a function f, set f\left(x\right)=y and solve the equation for x. The unique solution of the equation x=g\left(y\right) is the expression of the inverse of the function f.
In our problem:
f\left(x\right)=y
4\sqrt[5]{2x+3}-7=y
Solve this equation for x :
4\sqrt[5]{2x+3}=y+7
\sqrt[5]{2x+3}=\dfrac{y+7}{4}
2x+3=\left( \dfrac{y+7}{4} \right)^{5}
2x=\left( \dfrac{y+7}{4} \right)^{5}-3
x=\dfrac{1}{2}\left[ \left( \dfrac{y+7}{4} \right)^{5}-3 \right]
We can conclude that the inverse of the function f is:
g\left(y\right)=\dfrac{1}{2}\left[ \left( \dfrac{y+7}{4} \right)^{5}-3 \right] where y\in\mathbb{R}
Replacing y with x leads to:
g\left(x\right)=\dfrac{1}{2}\left[ \left( \dfrac{x+7}{4} \right)^{5}-3 \right] where x\in\mathbb{R}.
g is defined for x\in\mathbb{R}, and g\left(y\right)=\dfrac{1}{2}\left[ \left( \dfrac{y+7}{4} \right)^{5}-3 \right].