Let $f: N \rightarrow R: f(x)=4 x^{2}+12 x+15 .$ Show that $f: N \rightarrow$ range $(f)$ is invertible. Find $f^{-1} .$
To Show: that $\mathrm{f}$ is invertible
To Find: Inverse of $f$
[NOTE: Any functions is invertible if and only if it is bijective functions (i.e. one-one and onto)]
one-one function: A function $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ is said to be a one-one function or injective mapping if different
elements of A have different images in B. Thus for $x_{1}, x_{2} \in A \& f\left(x_{1}\right), f\left(x_{2}\right) \in B, f\left(x_{1}\right)=f\left(x_{2}\right) \leftrightarrow x_{1}=x_{2}$ or $x_{1} \neq$ $x_{2} \leftrightarrow f\left(x_{1}\right) \neq f\left(x_{2}\right)$
onto function: If range $=$ co-domain then $f(x)$ is onto functions.
So, We need to prove that the given function is one-one and onto.
Let $x_{1}, x_{2} \in R$ and $f(x)=4 x^{2}+12 x+15$ So $f\left(x_{1}\right)=f\left(x_{2}\right) \rightarrow\left(4 x_{1}^{2}+12 x_{1}+15\right)=\left(4 x_{2}^{2}+12 x_{2}+15\right)$, on solving we get $\rightarrow \mathrm{x}_{1}=\mathrm{x}_{2}$
So $f\left(x_{1}\right)=f\left(x_{2}\right) \leftrightarrow x_{1}=x_{2}, f(x)$ is one-one
Given co-domain of $f(x)$ is Range( $f$ ).
Let $y=f(x)=4 x^{2}+12 x+15$, So $x=\frac{-3+\sqrt{y-6}}{2}$ [Range of $f(x)=$ Domain of $y$ ]
So Domain of $y=$ Range of $f(x)=[6, \infty]$
Hence, Range of $f(x)=$ co-domain of $f(x)=[6, \infty]$
So, $f(x)$ is onto function
As it is bijective function. So it is invertible
Invers of $f(x)$ is $f^{-1}(y)=\frac{-3+\sqrt{y-6}}{2}$