Functions

Functions

3.1 Domain & Range of functions

A) State the suitable domain for each of the following

1. $f(x)=\sqrt{x-4}$

2. $f(x)=\sqrt{6-2x}$

B) Determine the range of each of the following functions

1. $f:x\mapsto x+4$, $0<x<5$

2. $f:x\mapsto x^{2}+7$,$x\in \Re$

3. $f:x\mapsto 2x-3$,$2<x \le 6$

4. $f:x\mapsto x^{2}-6x$,$0\le x\le 6$

5. $f :x\mapsto 3\sqrt{x}-4$, $x>0$

6. $f:x\mapsto \sqrt{3x-2}$,$2\le x\le 9$

7. $f :x\mapsto (x-3)^{2}+5$, $4\le x\le 6$

8. $f :x\mapsto (x-3)^{2}+5$, $2\le x\le 6$

C) Use the method of completing the square to find the range of each of these functions

1. $f(x)=x^{2}-6x+10$,$x\in \Re$

2. $f(x)=x^{2}+7x+1$,$x\in \Re$

3. $f(x)=x^{2}-3x+4$,$x\in \Re$

4. $f(x)=2x^{2}+8x+1$,$x\in \Re$

5. $f(x)=3x^{2}+5x-12$,$x\in \Re$

6. $f(x)=-x^{2}-6x+12$,$x\in \Re$

3.2 One-to-one & Many-to-one Functions

Determine which of the following functions are one-to-one and which are many-to-one

1. $f(x)=x+3$,$x\in \Re$

2. $f(x)=x^{2}+3$,$x\in \Re$

3. $f(x)=\frac{1}{x}$,$x\in \Re$,$x\neq 0$

4. $f(x)=(x-4)^{2}$,$2\le x\le 6$

5. $f(x)=x^{2}-4x$,$0<x<4$

6. $f(x)=x^{2}-4x$,$0<x<2$

7. $f(x)=\frac{2}{x-3}$,$-1<x<2$

3.3 Composite function

Given that $f(x)=2x+1$,$g(x)=x^{2}$ and $h(x)=\frac{1}{x}$ evaluate each of the following

1. $f(3)$

2. $g(2)$

3. $hg(2)$

4. $fg(-3)$

5. $gf(1)$

6. $gh(-2)$

7. $hf(4)$

8. $ff(5)$

9. $fgh(2)$

10. $hfg(4)$

The following $f$,$g$ and $h$ are defined by $f(x)=x^{2}$, $g(x)=\frac{3}{x}$ and $h(x)=2-x$. Find an expression for each of the following composite functions in term of $x$.

11. $gh$

12. $hg$

13. $g^{2}$

14. $h^{2}$

15. $ghf$

16. $hgf$

17. Given that $f(x)=x-3$,$g(x)=10x$ and $h(x)=\frac{1}{x}$, $x\neq 0$

     a) Find the expression of $fgh(x)$

     b) solve the equation $fgh(x)=x$

18. Funtions $f$ and $g$ are defined by $f(x)=3x+5$ and $g(x)=\frac{x-5}{3}$

     a) Find the expressions of $f^{2}(x)$ and $g^{2}(x)$

     b) Show that $fg(x)=x$ and $ggf(x)=x$

19. The functions $f$ and $g$ are defined by $f:x\mapsto \frac{x^{2}}{4}$ for {$x\in \Re$} and $g:x\mapsto \frac{2}{x-1}$ for {$x\in \Re$,$x\neq 1$}. Find expressions for the following case stating any values of $x$ for which the composite functions are undefined

(a) $fg(x)$

(b) $gf(x)$

(c) $f^{2}(x)$

(d) $g^{2}(x)$

3.4 Inverse Functions

A) Find the inverse of each of the following functions

1. $f:x\mapsto 3x+2$,$x\in \Re$

2. $f:x\mapsto 5x-1$,$x\in \Re$

3. $f:x\mapsto 4-3x$,$x\in \Re$

4. $f:x\mapsto \frac{2}{x}$,$x\neq 0$

B) Find the inverse of each of the following functions, and state the domain on which each inverse is defined

1. $f:x\mapsto x^{2}$,$x>2$

2. $f:x\mapsto \frac{1}{2+x}$,$x>0$

3. $f:x\mapsto \sqrt{x-2}$,$x>3$

4. $f:x\mapsto 3x^{2}-1$,$1<x<4$

5. $f:x\mapsto \frac{1}{x-2}$,$x>3$

6. $f:x\mapsto \sqrt{3-x}$,$x\le 2$

7. $f:x\mapsto (x-3)^{2}+5$,$4\le x\le 6$

8. $f:x\mapsto 5-\sqrt{x+3}$,$x\ge -3$

C)

1. Given that $f:x\mapsto 3x-4$,$x\in \Re$

     a) Find an expression for the inverse function $f^{-1}(x)$

     b) Sketch the graphs of $f(x)$ and $f^{-1}(x)$ on the same set of axes

     c) Solve the equation $f(x)=f^{-1}(x)$

2. A function is defined by $f(x)= x^{2}-6$,$x>0$

     a) Find an expression for the inverse function $f^{-1}(x)$

     b) Sketch the graphs of $f(x)$ and $f^{-1}(x)$ on the same set of axes

     c) Solve the equation $f(x)=f^{-1}(x)$

3. The function $g(x)=\frac{x+1}{x-3}$ is defined for $x\in \Re$,$x\neq 3$. Find its inverse function, and state the value of $x$ for which $g^{-1}(x)$ is not defined.

4. Functions $f$ and $g$ are defined by

$f:x\mapsto 3x+2$,$x\in \Re$;$g:x\mapsto \frac{6}{2x+3}$,$x\in \Re$,$x\neq -1.5$

     i) Find the value of $x$ for which $fg(x)=3$

     ii) Sketch, in a single diagram, the graphs of $y=f(x)$ and $y=f^{-1}(x)$, making clear the relationship between these two graphs

     iii) Express each of $f^{-1}(x)$ and $g^{-1}(x)$ in terms of $x$, and solve the equation $f^{-1}(x)=g^{-1}(x)$

3.5 Mixed Exercise

1. The linear function $f:x\mapsto ax+b$ is such that $f(1)=8$ and $f(3)=14$. Find $a$ and $b$

2. The linear function $f:x\mapsto ax+b$ where $a$ and $b$ are constants. Given that $f(2)=3$ and $f(-3)=13$, find $a$ and $b$. Hence, calculate the value of $c$ for which $f(c)=0$.

3. The function $m$ is defined by $m:x\mapsto 2+5x$, for $x\ge 1$. Find the range of $m$. Find the inverse of $m$, and find its domain and range.

4. Given that $f(x)=x^{2}+3$, and $g(x)=2x+a$ and $fg(x)=4x^{2}-8x+7$, find the value of the constant $a$.

5. The functions $f$ and $g$ are defined as follows:

$$f:x\mapsto x+1,x\in \Re$$ and $$g:x\mapsto x^{2}-3,x\in \Re$$ .

     i) Show that $fg(x)+gf(x)=2x^{2}+2x-4$

     ii) Hence solve $fg(x)+gf(x)=0$

6. The functions $g$ is defined by $g(x)=2x^{2}-3,x\ge 0$

     i) State the range of $g$ and sketch its graph

     ii) Explain why the inverse function $g^{-1}$ exist and sketch its graph

     iii) Given also that $h$ is defined by $h(x)=\sqrt{5x+2},x\ge -\frac{2}{5}$, solve the inequality $gh(x)\ge x$

7. The function $g$ is defined by $g:x\mapsto \frac{ax-4}{x+1},x\in \Re ,x\neq 1$. Find the positive value of $a$ for which there is only one value of $x$ satisfying $g(x)=x$

3.6 More exercise

1. i) Express $2x^{2}+8x-10$ in the form $a(x+b)^{2}+c$.

ii) For the curve $y=2x^{2}+8x-10$, state the least value of $y$ and the corresponding value of $x$

iii) Find the set of values of $x$ for which $y\ge 14$

     Given that $f:x\mapsto 2x^{2}+8x-10$ for the domain $x\ge k$

     iv) find the least value of $k$ for which $f$ is one-one.

     v) express $f^{-1}$ in terms of $x$ in this case

2. The equation of a curve is $y=8x-x^{2}$

     i) Express $8x-x^{2}$ in the form $a-(x+b)^{2}$, stating the numerical values of $a$ and $b$

     ii) Hence, or otherwise, find the coordinates of the stationary point of the curve

     iii) Find the set of values of $x$ for which $y\ge -20$

     The function $g$ is defined by $g:x\mapsto 8x-x^{2}$, for $x\ge 4$

     iv) State the domain and range of $g^{-1}$

     v) Find an expression, in terms of $x$, for $g^{-1}(x)$

3. Fundtions $f$ and $g$ are defined by

$$f:x\mapsto 2x-5,x\in \Re$$   $$g:x\mapsto \frac{4}{2-x},x\in \Re, x\neq 2$$

     i) Find the value of $x$ for which $fg(x)=7$

     ii) Express each of $f^{-1}(x)$ and $g^{-1}(x)$ in terms of $x$

     iii) Show that the equation $f^{-1}(x)=g^{-1}(x)$ has no real roots

     iv) Sketch, on a single diagram, the graphs of $y=f(x)$ and $y=f^{-1}(x)$, making clear the relationship between these two graphs.

4. The functions $f$ and $g$ are defined as follows:

$$f:x\mapsto x^{2}-2x,x\in \Re$$   $$g:x\mapsto 2x+3,x\in \Re$$

     i) Find the set of values of $x$ for which $f(x)>15$

     ii) Find the range of $f$ and state whether $f$ has an inverse

     iii) Show that the equation $gf(x)=0$ has no real solutions

     iv) Sketch, in a single diagram, the graphs of $y=g(x)$ and $y=g^{-1}(x)$, making the relationship between the graphs

5. The function $f:x\mapsto 2x-a$, where $a$ is a constant, is defined for all real $x$

     i) In the case where $a=3$, solve the equation $ff(x)=11$

     The function $g:x\mapsto x^{2}-6x$ is defined for all real $x$

     ii) Find the value of $a$ for which the equation $f(x)=g(x)$ has exactly one real solution

     The function $h:x\mapsto x^{2}-6x$ is defined for the domain $x\ge 3$

     iii) Express $x^{2}-6x$ in the form $(x-p)^{2}-q$, where $p$ and $q$ are constants

     iv) Find an expression for $h^{-1}(x)$ and state the domain of $h^{-1}$

6. Functions $f$ and $g$ are defined by $f:x\mapsto k-x$ for $x\in \Re$, where $k$ is a constant, $g:x\mapsto \frac{9}{x+2}$ for $x\in \Re,x\neq -2$

     i) Find the values of $k$ for which the equation $f(x)=g(x)$ has two equal roots and solve the equation $f(x)=g(x)$ in these cases

     ii) Solve the equation $fg(x)=5$ when $k=6$

     iii) Express $g^{-1}(x)$ in terms of $x$

7. The function $f$ is defined by $f:x\mapsto x^{2}-3x$ for $x\in \Re$

     i) Find the set of values of $x$ for which $f(x)>4$

     ii) Express $f(x)$ in the form $(x-a)^{2}-b$, stating the values of $a$ and $b$

     iii) Write down the range of $f$

     iv) State, with a reason, whether $f$ has an inverse

     The function $g$ is defined by $g:x\mapsto x-3\sqrt{x}$ for $x\ge 0$

     v) Solve the equation $g(x)=10$

8. Determine the set of values of the constant $k$ for which the line $y=4x+k$ does not intersect the curve $y=x^{2}$