Mathematics - Functions Question with Solution | TestHub

Let $y=f(x)=(5-(x-8{)}^5{)}^1{ᐟ}^3$, then:

$$y^3=5-(x-8{)}^5\Rightarrow (x-8{)}^5=5-y^3\Rightarrow x=8+(5-y^3{)}^1{ᐟ}^5$$

Let $z=g(x)=8+(5-x^3{)}^1{ᐟ}^5$. Now,

$$f(g(x))=(5-((8+(5-x^3{)}^1{ᐟ}^5)-8{)}^5{)}^1{ᐟ}^3$$

$$=(5-((5-x^3{)}^1{ᐟ}^5{)}^5{)}^1{ᐟ}^3=(5-(5-x^3){)}^1{ᐟ}^3=(x^3{)}^1{ᐟ}^3=x$$

Similarly, we can show that $g(f(x))=x$.

Hence, $g(x)=8+(5-x^3{)}^1{ᐟ}^5$ is the inverse of $f(x)$.