Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto {{x}^{2}}+cx+d,x\in \mat

Question

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto {{x}^{2}}+cx+d,x\in \mathbb{R},$ where $c$ and $d$ are constants, $\mathrm{g}:x\mapsto \ln x,x\in \mathbb{R},x>0$. Given that the composite function $\mathrm{gf}$ exists, find an inequality involving $c$ and $d$. [2] For the rest of the question, take $c=2,d=4$. Write down an expression for $\mathrm{gf}\left( x \right)$ and find exactly the range of $\mathrm{gf}$. [3] The graph of a function $\mathrm{h}$ is symmetrical about the line $x=k$ if $\mathrm{h}\left( x+k \right)=\mathrm{h}\left( -x+k \right)$ for all valid values of $x$. Using this definition, show that the graph of $\mathrm{gf}$ is symmetrical about the line $x=-1$. [2]

Accepted Answer

(i) $4d>c^2$; (ii) $gf(x)=\ln(x^2+2x+4)$, range $[\ln3,\infty)$; (iii) symmetric about $x=-1$

2020 DHS Promo Q3

2020 DHS Promo Q3

Video Solution:

Handwritten Solution: