Qudratics and Functions

Quadratics and Functions

Quadratics and functions (Part1)

1. The function $f$ is such that $f(x)=2\sin^{2}x-3\cos^{2}x$ for $0 \le x \le \pi$.

     i) Express $f(x)$ in the form  $a+b\cos^{2}x$, stating the values of $a$ and $b$.

     ii) State the greatest and least values of $f(x)$

     iii) Solve the equation $f(x)+1=0$

2. The function $f$ is defined $f:x\mapsto 2x^{2}-12x+7$ for $x\in \Re$

     i) Express $f(x)$ in the form $a(x-b)^{2}-c$

     ii) State the range of $f$

     iii) Find the set of values of $x$ for which $f(x)<21$

     The function $g$ is defined by $g:x\mapsto 2x+k$ for $x\in \Re$

     iv) Find the value of the constant $k$ for which the equation $gf(x)=0$ has two equal roots.

3. The function $f$ and $g$ are defined for $x\in \Re$ by $$f:x\mapsto 4x-2x^{2}$$ $$g:x\mapsto 5x+3$$

     i) Find the range of $f$

     ii) Find the value of the constant $k$ for which the equation $gf(x)=k$ has equal roots

4. The function $f:x\mapsto a+b \cos x$ is defined for $0 \le x \le 2\pi$. Given that $f(0)=10$ and that $f(\frac{2}{3}\pi)=1$, find

     i) the values of $a$ and $b$

     ii) the range of $f$

     iii) the exact value of $f(\frac{5}{6}\pi)$

5. The function $f:x\mapsto 2x^{2}-8x+14$ is defined for $x\in \Re$.

      i) Find the values of the constant $k$ for which the line $y+kx=12$ is tangent to the curve $y=f(x)$

     ii) Express $f(x)$ in the form $a(x+b)^{2}+c$, where $a$,$b$ and $c$ are constant

     iii) Find the range of $f$

     The function $g:x\mapsto 2x^{2}-8x+14$ is defined for $x \ge A$

     iv) Find the smallest value of A for which $g$ has an inverse.

     v) For this value of A, find an expression for $g^{-1}(x)$ in term of $x$

6. Functions $f$ and $g$ are defined for $x\in \Re$ by $$f:x\mapsto 2x+3$$ $$g:x\mapsto x^{2}-2x$$ Express $gf(x)$ in the form $a(x+b)^{2}+c$, where $a$,$b$ and $c$ are constant.

7. A function $f$ is defined by $f:x\mapsto 3-2\tan (\frac{1}{2}x)$ for $0\le x < \pi$

     i) State the range of $f$

     ii) State the exact value of $f(\frac{2}{3}\pi)$

     iii) Sketch the graph of $y=f(x)$

     iv) Obtain an express, in terms of $x$, for $f^{-1}(x)$

8. A curve has equation $y=kx^{2}+1$ and a line has equation $y=kx$, where $k$ is a non-zero constant.

     i) Find the set of values of $k$ for which the curve and the line have no common points.

     ii) State the value of $k$ for which the line is a tangent to the curve and, for this case, find the coordinates of the point where the line touches the curve.

9.  The function $f$ is defined by $f(x)=x^{2}-4x+7$ for $x>2$

     i) Express $f(x)$ in the form $(x-a)^{2}+b$ and hence state the range of $f$

     ii) Obtain an expression for $f^{-1}(x)$ and state the domain of $f^{-1}$

     The function $g$ is defined by $g(x)=x-2$ for $x>2$

     The function $h$ is such that $f=hg$ and the domain of $h$ is $x>0$

     iii) Obtain an expression for $h(x)$

10. The diagram shows the function $f$ defined for $0\le x \le 6$ by

$$x\mapsto \frac{1}{2}x^{2} \text{ for }0 \le x \le 2$$,

$$x\mapsto \frac{1}{2}x +1 \text{ for } 2<x\le 6$$

     i) State the range of $f$

     ii) Copy the diagram and on your copy sketch the graph of $y=f^{-1}(x)$

     iii) Obtain expression to defined $f^{-1}(x)$, giving the set of values of $x$ for which each expression is valid

11. Functions $f$ and $g$ are defined for $x\in \Re$ by

$$f:x\mapsto 2x-1$$ $$g:x\mapsto x^{2}-2$$

     i) Find and simplify expressions for $fg(x)$ and $gf(x)$

     ii) Hence find the value of $a$ for which $fg(a)=gf(a)$

     iii) Find the value of $b(b\neq a)$ for which $g(b)=b$

     iv) Find and simplify an expression for $f^{-1}g(x)$

     The function $h$ is defined by $h:x\mapsto x^{2}-2$, for $x\le 0$

     v) Find an expression for $h^{-1}(x)$

12. The function $f$ is such that $f(x)=3-4\cos ^(k)x$, for $0\le x \le \pi$, where $k$ is a constant

     i) In the case where $k=2$,

            a) Find the range of $f$

            b) Find the exact solutions of the equation $f(x)=1$

     ii) In the case where $k=1$

            a) Sketch the graph of $y=f(x)$

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

Quadratics and functions(part 2)

1. Functions $f$ and $g$ are defined by $f:x\mapsto 3x-4$, $x\in \Re$, $g:x\mapsto 2(x-1)^{3}+8$, $x>1$

     i) Evaluate $fg(2)$

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

     iii) Obtain an expression for $g’(x)$ and use your answer to explain why $g$ has an inverse

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

2. Functions $f$ and $g$ are defined by $f:x\mapsto 2x^{2}-8x+10$ for $0\le x\le 2$, $g:x\mapsto x$ for $0\le x\le 10$

     i) Express $f(x)$ in the form of $a(x+b)^{2}+c$, where $a$, $b$ and $c$ are constants

     ii) State the range of $f$

     iii) State the domain of $f^{-1}(x)$

     iv) Sketch on the same diagram the graphs of $y=f(x)$, $y=g(x)$ and $y=f^{-1}(x)$, making clear the relationship between the graphs

     v) Find an expression for $f^{-1}(x)$

3. The functions $f$ and $g$ are defined for $x\in \Re$ by $$f:x\mapsto 3x+a$$ $$g;x\mapsto b-2x$$ where $a$ and $b$ are constants. Given that $ff(2)=10$ and $g^{-1}(2)=3$, find

     i) the values of $a$ and $b$

     ii) an expression for $fg(x)$

4. i) Sketch, on the same diagram, the graphs of $y=\sin x$ and $y=\cos 2x$ for $0^{\circ}\le x \le 180^{\circ}$

ii) Verify that $x=30^{\circ}$ is a root of the equation $\sin x=\cos 2x$,  and state the other root of this equation for which for $0^{\circ}\le x \le 180^{\circ}$

 iii) Hence state the set of values of $x$, for for $0^{\circ}\le x \le 180^{\circ}$, for which $\sin x <\cos 2x$

5. The function $f:x\mapsto x^{2}-4x+k$ is defined for the domain $x\ge p$, where $k$ and $p$ are constants

     i) Express $f(x)$ in the form $(x+a)^{2}+b+k$, where $a$ and $b$ are constants

     ii) State the range of $f$ in terms of $k$

     iii) State the smallest value of $p$ for which $f$ is one-to-one

     iv) For the value of $p$ found in part iii), find an expression for $f^{-1}(x)$ and state the domain of $f^{-1}$, giving your answers in term of $k$

6. Functions $f$ and $g$ are defined by $f:x\mapsto 2x+5$ for $x\in \Re$, $g:x\mapsto \frac{8}{x-3}$ for $x\in \Re$, $x\neq 3$

     i) Obtain expressions, in term of $x$, for $f^{-1}(x)$ and $g^{-1}(x)$, stating the value of $x$ for which $g^{-1}(x)$ is not defined

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

     iii) Given that the equation $fg(x)=5-kx$, where $k$ is a constant, has no solutions, find the set of possible values of $k$

7. The function $f$ is such that $f(x)=8-(x-2)^{2}$, for $x\in \Re$

     i) Find the coordinates and the nature of the stationary point on the curve $y=f(x)$

     The function $g$ is such that $g(x)=8-(x-2)^{2}$, for $k\le x\le 4$, where $k$ is a constant

     ii) State the smallest value of $k$ for which $g$ has an inverse

     For this value of $k$

     iii) Find an expression for $g^{-1}(x)$

     iv) Sketch, on the same diagram, the graphs of $y=g(x)$ and $y=g^{-1}(x)$

8. The function $f$ is defined by $f(x)=4x^{2}-24x+11$, for $x\in \Re$

     i) Express $f(x)$ in the form $a(x-b)^{2}+c$ and hence state the coordinates of the vertex of the graph of $y=f(x)$

     The function $g$ is defined by $g(x) )=4x^{2}-24x+11$, for $x\le 1$

     ii) State the range of $g$

     iii) Find the expression for $g^{-1}(x)$ and state the domain of $g^{-1}$

9. A function $f$ is such the $f(x)=\sqrt{\frac{x+3}{2}}+1$, for $x\ge -3$

     i) Find $f^{-1}(x)$ in the form $ax^{2}+bx+c$, where $a$, $b$ and $c$ are constants

     ii) Find the domain of $f^{-1}$

10.

     i) The diagram shows part of the curve $y=11-x^{2}$ and part of straight line $y=5-x$ meeting at the point $A(p,q)$, where $p$ and $q$ are positive constants. Find the values of $p$ and $q$

      ii) The function $f$ is defined for the domain $x\ge 0$ by $$f(x)=\begin{cases}11-x^{2}, & \text{for $0\le x\le p$}\\5-x, &\text{for $x>p$}\end{cases}$$ Express $f^{-1}(x)$ in a similar way.