The square root function and the inverse function

Calculus, grade 10 (5 units) Lesson 18 of 26Learning

The square root function and the inverse function

The inverse function

The inverse function of $\displaystyle f$ goes the way back: if $\displaystyle f$ gives $\displaystyle y$ for $\displaystyle x$ , the inverse gives $\displaystyle x$ for $\displaystyle y$ .

$\displaystyle {f(x)}={2x+1}$ gives 7 for 3, and the inverse $\displaystyle {g(x)}={\frac{x-1}{2}}$ gives 3 for 7.
To find the inverse: write $\displaystyle {y}={f(x)}$ , get $\displaystyle x$ on its own, and swap the names of the variables.
On a graph, swapping $\displaystyle x$ and $\displaystyle y$ is a reflection in the line $\displaystyle {y}={x}$ : the point $\displaystyle (3,7)$ goes to $\displaystyle (7,3)$ .

\displaystyle f(x)=2x+1

\displaystyle g(x)=\frac{x-1}{2}

\displaystyle y=x

A function has an inverse only if each value of $\displaystyle y$ comes from one $\displaystyle x$ alone – for example a function that only rises or only falls. $\displaystyle {y}={x^2}$ has no inverse: both $\displaystyle {x}={2}$ and $\displaystyle {x}={-2}$ give $\displaystyle {y}={4}$ , and there is no single way back from 4.

The square root function

Looked at only where $\displaystyle x\geq0$ , $\displaystyle {y}={x^2}$ rises, and it has an inverse: the square root function $\displaystyle {y}={\sqrt{x}}$ . $\displaystyle \sqrt{a}$ is the non-negative number whose square is $\displaystyle a$ .

\displaystyle \sqrt{x}

\displaystyle =

\displaystyle x^{\frac{1}{2}}

On the graph below, $\displaystyle {y}={\sqrt{x}}$ is the reflection of the right half of the parabola in the line $\displaystyle {y}={x}$ :

\displaystyle f(x)=\sqrt{x}

\displaystyle y=x^{2}

\displaystyle y=x

The domain: $\displaystyle x\geq0$ , and always $\displaystyle \sqrt{x}\geq0$ .
Its only zero: $\displaystyle (0,0)$ .
It rises in all its domain, and $\displaystyle (0,0)$ is a minimum point at the end of the domain.
It rises more and more slowly: from 0 to 1 it rises by 1, and from 1 to 4 – also only by 1.

The cube root and the root of a square

$\displaystyle {y}={x^3}$ rises everywhere, so it has an inverse on the whole line: the cube root function $\displaystyle {y}={\sqrt[3]{x}}$ .

\displaystyle \sqrt[3]{x}

\displaystyle =

\displaystyle x^{\frac{1}{3}}

It is defined for every $\displaystyle x$ : $\displaystyle {\sqrt[3]{8}}={2}$ , and $\displaystyle {\sqrt[3]{-8}}={-2}$ because $\displaystyle {(-2)^3}={-8}$ .

The square root of a square needs care: $\displaystyle {\sqrt{(-3)^2}}={\sqrt{9}}={3}$ , not $\displaystyle -3$ . A square root is never negative, so:

\displaystyle \sqrt{x^2}

\displaystyle =

\displaystyle |x|

Shifts and stretches of

\displaystyle \sqrt{x}

The shifts and stretches you have learnt work on $\displaystyle \sqrt{x}$ too:

\displaystyle y

\displaystyle =

\displaystyle a\sqrt{x-p}+k

The graph starts at the point $\displaystyle (p,k)$ , and the domain is $\displaystyle x\geq p$ .
$\displaystyle a>0$ : the graph rises from that point. $\displaystyle a<0$ : it falls from it.

On the graph below, $\displaystyle {y}={2\sqrt{x-1}+1}$ :

\displaystyle f(x)=\sqrt{x}

\displaystyle g(x)=2\sqrt{x-1}+1

\displaystyle A=(1,1)

$\displaystyle \sqrt{x}$ was shifted one unit right and one unit up, and stretched by 2.
It starts at $\displaystyle (1,1)$ , is defined where $\displaystyle x\geq1$ and rises.

The root of a function

In $\displaystyle {y}={\sqrt{f(x)}}$ the inner function is $\displaystyle f$ , and the outer one is the root. What you read from $\displaystyle f$ :

The domain: a negative number has no square root, so $\displaystyle \sqrt{f(x)}$ is defined where $\displaystyle f(x)\geq0$ .
The zeros: the zeros of $\displaystyle f$ .
Rising and falling: the root rises, so $\displaystyle \sqrt{f(x)}$ rises where $\displaystyle f$ rises, and falls where $\displaystyle f$ falls.

The example below reads $\displaystyle {y}={\sqrt{6-2x}}$ this way:

$\displaystyle 6-2x\geq0$ where $\displaystyle x\leq3$ : that is the domain.
$\displaystyle {6-2x}={0}$ where $\displaystyle {x}={3}$ : the zero $\displaystyle (3,0)$ , at the end of the domain.
$\displaystyle 6-2x$ falls, so $\displaystyle \sqrt{6-2x}$ falls in all its domain too.

An equation with a root

An equation with a root is solved by squaring both sides. Squaring can add a solution: $\displaystyle {2}={-2}$ , which is false, gives $\displaystyle {4}={4}$ once squared. So check every solution in the original equation.

The example below solves $\displaystyle {\sqrt{x+3}}={x-3}$ :

Squaring: $\displaystyle {x+3}={x^2-6x+9}$ , so $\displaystyle {x^2-7x+6}={0}$ , that is $\displaystyle {(x-1)(x-6)}={0}$ .
$\displaystyle {x}={1}$ : the left side is $\displaystyle {\sqrt{4}}={2}$ , the right side $\displaystyle {1-3}={-2}$ . Not a solution.
$\displaystyle {x}={6}$ : the left side is $\displaystyle {\sqrt{9}}={3}$ , the right side $\displaystyle {6-3}={3}$ . The only solution.

Exercise 1

The function is $\displaystyle {f(x)}={3x-6}$ .
Find its inverse function, and write it in a row of your own on the board.
Check that its graph is the reflection of the graph of $\displaystyle f$ in the line $\displaystyle {y}={x}$ .

\displaystyle f(x)=3x-6

\displaystyle y=x

Exercise 2

The board shows the graph of $\displaystyle {f(x)}={\sqrt{x}}$ , and three graphs made from it by shifts, stretches and reflections.
Write a function of the form $\displaystyle {y}={a\sqrt{x-p}+k}$ for each of them.

\displaystyle f(x)=\sqrt{x}

\displaystyle g(x)=\,?

\displaystyle h(x)=\,?

\displaystyle k(x)=\,?

Exercise 3

Find the domain of each function.

$\displaystyle {f(x)}={\sqrt{8-2x}}$

$\displaystyle {g(x)}={\sqrt{x^2-x-6}}$

Write the zeros of $\displaystyle f$ and of $\displaystyle g$ .
Where does $\displaystyle g$ rise? Explain by the parabola $\displaystyle {y}={x^2-x-6}$ .

Exercise 4

Solve the equations.
Check every solution in the original equation.

Exercise 5

Work out.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Dana wrote: “ $\displaystyle {\sqrt{x^2}}={x}$ for every $\displaystyle x$ ”.
Give an example that shows she is wrong, and write what is true.

Formulas: calculus, grade 10 (5 units)

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Algebra

The square of a sum and of a difference
$\displaystyle (a\pm b)^2$ $\displaystyle =$ $\displaystyle a^2\pm2ab+b^2$
The difference of two squares
$\displaystyle a^2-b^2$ $\displaystyle =$ $\displaystyle (a-b)(a+b)$
The cube of a sum and of a difference
$\displaystyle (a\pm b)^3$ $\displaystyle =$ $\displaystyle a^3\pm3a^2b+3ab^2\pm b^3$
The sum and the difference of two cubes
$\displaystyle a^3\pm b^3$ $\displaystyle =$ $\displaystyle (a\pm b)(a^2\mp ab+b^2)$
The quadratic formula for $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Factoring a trinomial – $\displaystyle x_1$ , $\displaystyle x_2$ the roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle ax^2+bx+c$ $\displaystyle =$ $\displaystyle a(x-x_1)(x-x_2)$

The properties of a function

An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transformations of a graph

A shift of $\displaystyle k$ units up ( $\displaystyle k<0$ – down)
$\displaystyle y=f(x)+k$
A shift of $\displaystyle p$ units right ( $\displaystyle p<0$ – left)
$\displaystyle y=f(x-p)$
A vertical stretch ( $\displaystyle |a|>1$ ) or squeeze ( $\displaystyle |a|<1$ )
$\displaystyle y=a\cdot f(x)$
A horizontal squeeze by $\displaystyle a$ ( $\displaystyle a>1$ ) or stretch ( $\displaystyle 0<a<1$ )
$\displaystyle y=f(ax)$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$

The derivative

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the secant
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
The derivative at a point – the slope of the tangent
$\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}$
The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a constant
$\displaystyle (c)'$ $\displaystyle =$ $\displaystyle 0$
The derivative of a constant multiple
$\displaystyle (c\cdot f(x))'$ $\displaystyle =$ $\displaystyle c\cdot f'(x)$
The derivative of a sum and of a difference
$\displaystyle (f(x)\pm g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\pm g'(x)$
The instantaneous speed – the derivative of the distance with respect to time
$\displaystyle v(t)$ $\displaystyle =$ $\displaystyle s'(t)$

Rules of differentiation

The derivative of a product of functions
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient of functions
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
The derivative of a composite function – the outer derivative times the inner derivative
$\displaystyle [f(u(x))]'$ $\displaystyle =$ $\displaystyle f'(u)\cdot u'(x)$
The derivative of $\displaystyle \frac{1}{x}$
$\displaystyle \left(\frac{1}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{1}{x^2}$
The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

The tangent to a graph

The slope of the tangent at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle m=f'(x_1)$
The equation of the tangent
$\displaystyle y-y_1=m(x-x_1)$

Rising, falling and extremum points

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A point that may be an extremum
$\displaystyle f'(x)=0$

Asymptotes

A vertical asymptote – $\displaystyle a$ a zero of the denominator that is not a zero of the numerator
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle b$ the value the function approaches as $\displaystyle x$ grows or falls without end
$\displaystyle y=b$