Reciprocal and Square Transformations

Welcome, students! In this lesson, you will explore two important types of function transformations in IB Mathematics: Analysis and Approaches HL: reciprocal and square transformations. These ideas help you understand how graphs change when you apply operations such as taking the reciprocal $\frac{1}{f(x)}$ or squaring a function $[f(x)]^2$. They appear in many parts of mathematics, including rational functions, radical functions, coordinate geometry, and modelling real-world situations like speed, area, intensity, and rates ⚡.

Lesson Objectives

By the end of this lesson, you should be able to:

explain the meaning of reciprocal and square transformations in function language,
sketch and interpret graphs after these transformations,
identify domain and range changes caused by $\frac{1}{f(x)}$ and $[f(x)]^2$,
connect these ideas to asymptotes, zeros, and symmetry,
solve equations and inequalities involving transformed functions,
use correct IB-style reasoning when describing function behaviour.

1. What Do Reciprocal and Square Transformations Mean?

In function notation, a transformation changes a function’s graph in a predictable way. If a function is $f(x)$, then a reciprocal transformation creates a new function such as $g(x)=\frac{1}{f(x)}$. A square transformation creates a new function such as $h(x)=[f(x)]^2$.

These are not the same as simple shifts or stretches. Instead, they change the output of the function itself. That means we are transforming the $y$-values, not the $x$-values. This is sometimes called a transformation on the output or a “mapping” of the function values.

For students, a useful way to think about it is this:

if $f(x)$ gives a height, then $\frac{1}{f(x)}$ gives a reciprocal height,
if $f(x)$ gives a quantity, then $[f(x)]^2$ gives the square of that quantity.

These transformations are especially important when studying composite functions and inverses because they involve how function values are used and interpreted.

2. Reciprocal Transformations: $g(x)=\frac{1}{f(x)}$

A reciprocal transformation turns each output value $f(x)$ into $\frac{1}{f(x)}$. This has several important consequences.

First, whenever $f(x)=0$, the new function $g(x)=\frac{1}{f(x)}$ is undefined. That means the zeros of $f(x)$ become vertical asymptotes or breaks in the graph of $g(x)$, depending on the function. This is a major feature of reciprocal transformations.

Second, values of $f(x)$ with large magnitude become small after taking the reciprocal. For example, if $f(x)=10$, then $g(x)=\frac{1}{10}=0.1$. If $f(x)=-2$, then $g(x)=\frac{1}{-2}=-\frac{1}{2}$. So large positive and negative outputs are pulled toward $0$, but they keep their sign.

Third, values near $0$ become very large in magnitude after the reciprocal. For instance, if $f(x)=0.1$, then $g(x)=10$. This explains why reciprocal graphs often shoot upward or downward near zeros of the original function.

Example 1: A simple linear function

Suppose $f(x)=x-2$. Then

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

The original function has a zero at $x=2$, so $g(x)$ is undefined there. Therefore, $x=2$ is a vertical asymptote. Also, as $x$ gets very large, $x-2$ becomes large, so $g(x)$ approaches $0$. This gives a horizontal asymptote at $y=0$.

This graph has two branches. For $x>2$, $x-2>0$, so $g(x)>0$. For $x<2$, $x-2<0$, so $g(x)<0$. The graph reflects the sign of the original function.

Example 2: A quadratic in the denominator

If $f(x)=x^2-1$, then

$$g(x)=\frac{1}{x^2-1}.$$

The zeros of $f(x)$ are $x=-1$ and $x=1$, so $g(x)$ is undefined at both values. That means there are vertical asymptotes at $x=-1$ and $x=1$. Since $x^2-1$ is positive for $|x|>1$ and negative for $|x|<1$, the reciprocal changes sign in those intervals too.

This kind of reasoning is common in IB questions because you are expected to connect algebraic features to graph behaviour.

3. Square Transformations: $h(x)=[f(x)]^2$

A square transformation changes each output value $f(x)$ into $[f(x)]^2$. Since squares are always non-negative, the new function satisfies

$$[f(x)]^2\ge 0$$

for all $x$ in the domain of $f$.

This has big implications:

negative values of $f(x)$ become positive after squaring,
zeros of $f(x)$ stay at zero,
the graph of $[f(x)]^2$ may “lift” parts of the original graph above the $x$-axis.

One important idea is that a square transformation removes sign information. If $f(x)=3$ and $f(x)=-3$, both give $[f(x)]^2=9$. So the transformed graph no longer tells you whether the original output was positive or negative. This loss of information matters in solving equations.

Example 3: Squaring a linear function

Suppose $f(x)=x-2$. Then

$$h(x)=(x-2)^2.$$

This is a parabola opening upward with vertex at $(2,0)$. The original function $f(x)$ crosses the $x$-axis at $x=2$, and that point remains a zero after squaring. But now the graph is always on or above the $x$-axis.

If we compare $f(x)$ and $[f(x)]^2$, we see that values between $-1$ and $1$ become smaller when squared, while values with magnitude greater than $1$ become larger. For example, if $f(x)=\frac{1}{2}$, then $[f(x)]^2=\frac{1}{4}$, but if $f(x)=-3$, then $[f(x)]^2=9$.

Example 4: Squaring a quadratic

If $f(x)=x^2-4$, then

$$h(x)=(x^2-4)^2.$$

This new function has zeros where $x^2-4=0$, so at $x=\pm 2$. However, because the output is squared, the graph touches the $x$-axis at those points and stays non-negative everywhere.

4. Comparing Reciprocal and Square Transformations

These transformations behave very differently:

The reciprocal $\frac{1}{f(x)}$ is undefined where $f(x)=0$.
The square $[f(x)]^2$ is always defined wherever $f(x)$ is defined.
The reciprocal can be positive or negative depending on the sign of $f(x)$.
The square is never negative.
Reciprocal graphs often have asymptotes.
Square transformations often create a graph that sits above the $x$-axis.

A useful IB thinking strategy is to ask three questions about any transformed function:

Where is it defined?
Where are the zeros and asymptotes?
How do signs and magnitudes change?

This helps you move from algebra to graphing and back again.

5. Solving Equations and Inequalities Involving Transformed Functions

In IB Mathematics, you often need to solve equations such as

$$\frac{1}{f(x)}=k$$

$$[f(x)]^2=k.$$

Reciprocal equations

$$\frac{1}{f(x)}=k,$$

then, provided $k\ne 0$, you can rewrite this as

$$f(x)=\frac{1}{k}.$$

But you must also check that $f(x)\ne 0$ because the reciprocal is undefined at zero.

For example, if

$$\frac{1}{x-2}=3,$$

then

$$x-2=\frac{1}{3},$$

$$x=\frac{7}{3}.$$

Square equations

$$[f(x)]^2=k,$$

then you need $k\ge 0$ for real solutions. Then solve

$$f(x)=\pm\sqrt{k}.$$

This “plus or minus” step is essential.

For example, if

$$(x-2)^2=9,$$

then

$$x-2=\pm 3,$$

$$x=5 \quad \text{or} \quad x=-1.$$

Inequalities

For reciprocal and square transformations, inequalities require careful sign analysis.

If $\frac{1}{f(x)}>0$, then $f(x)>0$.
If $\frac{1}{f(x)}<0$, then $f(x)<0$.
If $[f(x)]^2\le 4$, then $-2\le f(x)\le 2$.

These are powerful because they let you translate a transformed inequality back into an easier inequality involving $f(x)$.

6. Why These Transformations Matter in the Wider Functions Topic

Reciprocal and square transformations connect directly to the broader study of functions in IB Mathematics: Analysis and Approaches HL. They build on ideas such as domain, range, graphing, composition, inverse functions, and algebraic manipulation.

They also appear in models. For example:

reciprocal relationships can describe quantities that vary inversely, such as time and speed over a fixed distance,
square relationships can describe area, intensity, or energy in certain settings.

Understanding these transformations improves your ability to interpret function notation, solve problems, and explain behaviour clearly using mathematical language.

Conclusion

students, reciprocal and square transformations are key tools for understanding how functions change. A reciprocal transformation $g(x)=\frac{1}{f(x)}$ creates asymptotes where $f(x)=0$ and flips signs while shrinking large values toward $0$. A square transformation $h(x)=[f(x)]^2$ removes negative outputs, keeps zeros, and always produces non-negative values. Both transformations are central to graph analysis, solving equations, and interpreting functions in IB Mathematics: Analysis and Approaches HL. Mastering them helps you move confidently between algebraic expressions, graphs, and real-world applications 📘.

Study Notes

Reciprocal transformation: $g(x)=\frac{1}{f(x)}$.
Square transformation: $h(x)=[f(x)]^2$.
If $f(x)=0$, then $\frac{1}{f(x)}$ is undefined.
Zeros of $f(x)$ often become vertical asymptotes for $\frac{1}{f(x)}$.
$[f(x)]^2\ge 0$ for all allowed $x$.
Squaring removes sign information: both $3$ and $-3$ become $9$.
To solve $[f(x)]^2=k$, use $f(x)=\pm\sqrt{k}$ when $k\ge 0$.
To solve $\frac{1}{f(x)}=k$, rewrite as $f(x)=\frac{1}{k}$ when $k\ne 0$.
Reciprocal graphs often have asymptotes; square-transformed graphs often stay on or above the $x$-axis.
Always check domain, range, zeros, and sign changes when studying transformed functions.