Composition of functions — practice questions

Question 1 · difficulty L3 · understanding

For some a, b, c $$\in\mathbb{N}$$, let $$f(x) = ax - 3$$ and $$\mathrm{g(x)=x^b+c,x\in\mathbb{R}}$$. If $${(fog)^{ - 1}}(x) = {\left( {{{x - 7} \over 2}} \right)^{1/3}}$$, then $$(fog)(ac) + (gof)(b)$$ is equal to ____________.

Question 2 · difficulty L3 · understanding

Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If $$f(g(x)) = 8{x^2} - 2x$$ and $$g(f(x)) = 4{x^2} + 6x + 1$$, then the value of $$f(2) + g(2)$$ is _________.

Question 3 · difficulty L3 · understanding

Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Define f : S $$\to$$ S as \begin{aligned}f(n) = \left\{ {\matrix{ {2n} & , & {if\,n = 1,2,3,4,5} \cr {2n - 11} & , & {if\,n = 6,7,8,9,10} \cr } } \right.\end{aligned}. Let g : S $$\to$$ S be a function such that \begin{aligned}fog(n) = \left\{ {\matrix{ {n + 1} & , & {if\,n\,\,is\,odd} \cr {n - 1} & , & {if\,n\,\,is\,even} \cr } } \right.\end{aligned}. Then $$g(10)g(1) + g(2) + g(3) + g(4) + g(5))$$ is equal to _____________.

Question 4 · difficulty L3 · understanding

Let $$f:R \to R$$ be a function defined by $$f(x) = {\left( {2\left( {1 - {{{x^{25}}} \over 2}} \right)(2 + {x^{25}})} \right)^{{1 \over {50}}}}$$. If the function $$g(x) = f(f(f(x))) + f(f(x))$$, then the greatest integer less than or equal to g(1) is ____________.

Question 5 · difficulty L3 · understanding

If $g(x)=3 x^2+2 x-3, f(0)=-3$ and $4 g(f(x))=3 x^2-32 x+72$, then $f(g(2))$ is equal to:

• A. $\frac{7}{2}$
• B. $-\frac{25}{6}$
• C. $\frac{25}{6}$
• D. $-\frac{7}{2}$

Question 6 · difficulty L3 · understanding

Let $f(x)=\log _{\mathrm{e}} x$ and $g(x)=\frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1}$. Then the domain of $f \circ g$ is

• A. $(0, \infty)$
• B. $[1, \infty)$
• C. $\mathbb{R}$
• D. $[0, \infty)$

Question 7 · difficulty L3 · understanding

Let $$f, g: \mathbf{R} \rightarrow \mathbf{R}$$ be defined as : $$f(x)=|x-1| \text { and } g(x)= \begin{cases}\mathrm{e}^x, & x \geq 0 \\ x+1, & x \leq 0 .\end{cases}$$ Then the function $$f(g(x))$$ is

• A. neither one-one nor onto.
• B. one-one but not onto.
• C. both one-one and onto.
• D. onto but not one-one.

Question 8 · difficulty L3 · understanding

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ and $g: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)=\left\{$$\begin{array}{ll}\log _{\mathrm{e}} x, & x>0 \\ \mathrm{e}^{-x}, & x \leq 0\end{array}$$\right.$and$g(x)=\left\{$$\begin{array}{ll}x, & x \geqslant 0 \\ \mathrm{e}^x, & x<0\end{array}$$\right.$. Then, gof :$\mathbf{R} \rightarrow \mathbf{R}$ is :

• A. one-one but not onto
• B. neither one-one nor onto
• C. onto but not one-one
• D. both one-one and onto

Question 9 · difficulty L3 · understanding

If $$f(x)=\frac{4 x+3}{6 x-4}, x \neq \frac{2}{3}$$ and $$(f \circ f)(x)=g(x)$$, where $$g: \mathbb{R}-\left\{\frac{2}{3}\right\} \rightarrow \mathbb{R}-\left\{\frac{2}{3}\right\}$$, then $$(g ogog)(4)$$ is equal to

• A. $$-4$$
• B. $$\frac{19}{20}$$
• C. $$-\frac{19}{20}$$
• D. 4

Question 10 · difficulty L3 · understanding

If $$f(x)=\left\{\begin{array}{cc}2+2 x, & -1 \leq x < 0 \\ 1-\frac{x}{3}, & 0 \leq x \leq 3\end{array} ; g(x)=\left\{\begin{array}{cc}-x, & -3 \leq x \leq 0 \\ x, & 0 < x \leq 1\end{array}\right.\right.$$, then range of $$(f o g)(x)$$ is

• A. $$[0,1)$$
• B. $$[0,3)$$
• C. $$(0,1]$$
• D. $$[0,1]$$