Explain in words why a given function does not have an inverse

Functions – Inverses (IGCSE Additional Mathematics 0606)

1. Key terminology and notation

• Function \(f\): a rule that assigns to each element of a set called the domain exactly one element of a set called the range.
• Domain of \(f\): the set of all permissible inputs \(x\).
  Example: for \(f(x)=\sqrt{x}\) the domain is \(\{x\in\mathbb{R}\mid x\ge 0\}\).
• Range of \(f\): the set of all values that actually occur as outputs.
  For the same \(f(x)=\sqrt{x}\) the range is \(\{y\in\mathbb{R}\mid y\ge 0\}\).
• One‑to‑one (injective): different inputs give different outputs. Formally, \(f(x_1)=f(x_2)\Rightarrow x_1=x_2\).
• Many‑one: at least two distinct inputs give the same output; the function is not one‑to‑one.
• Onto (surjective): every element of the stated range is produced by some input.
• Bijective: a function that is both one‑to‑one and onto. Only bijective functions have an inverse that is also a function.
• Inverse function \(f^{-1}\): a function that “undoes’’ \(f\); it satisfies \[ f^{-1}(f(x))=x\qquad\text{and}\qquad f(f^{-1}(y))=y . \]
• Function notation (IGCSE wording): \[ y=f(x),\qquad x=f^{-1}(y),\qquad (g\circ f)(x)=g\bigl(f(x)\bigr),\qquad f^2(x)=f\bigl(f(x)\bigr),\qquad fg(x)=f(x)g(x). \] Note: \(f^2(x)\) means \(f(f(x))\) and must **not** be used with trigonometric functions in the IGCSE exam.

2. Composite functions

Given two functions \(f\) and \(g\), the composite \(g\circ f\) is defined by \((g\circ f)(x)=g\bigl(f(x)\bigr)\). The order matters:

1. Let \(f(x)=x+2\) and \(g(x)=x^{2}\).
2. \((g\circ f)(x)=g(f(x))=(x+2)^{2}=x^{2}+4x+4.\)
3. \((f\circ g)(x)=f(g(x))=x^{2}+2.\)
4. Since \((g\circ f)(x)eq(f\circ g)(x)\), the order of composition is crucial.

When forming a composite, the domain is the set of all \(x\) for which both \(f(x)\) is defined and \(g\bigl(f(x)\bigr)\) is defined.

3. Absolute‑value transformation

For any function \(f\), the graph of \(y=|f(x)|\) is obtained by reflecting any part of the graph of \(y=f(x)\) that lies below the \(x\)-axis upwards, so that all \(y\)-values become non‑negative. The effect on different types of functions is summarised below.

4. When a function fails to have an inverse

For an inverse to exist the original function must be bijective. In the IGCSE syllabus the most common reason a function does not have an inverse is that it is not one‑to‑one (i.e. it is many‑one). In words, you should be able to state:

“The function does not have an inverse because it fails the horizontal‑line test (or, equivalently, because two different inputs give the same output).”

5. Horizontal line test (graphical check for one‑to‑one)

1. Draw any horizontal line \(y=c\) on the graph of \(y=f(x)\).
2. If the line meets the graph in more than one point, the function is not one‑to‑one.
3. Consequently the function cannot have an inverse that is itself a function.

6. Typical examples that lack inverses

7. Restoring an inverse – restricting domain or range

Many functions become bijective when we limit the set of inputs (or outputs). The restriction must make the function one‑to‑one and onto the new range.

Example 1 – Quadratic function

1. Original: \(f(x)=x^{2}\) with domain \(\mathbb{R}\). Not one‑to‑one.
2. Restrict domain to \(x\ge 0\) (or \(x\le 0\)).
3. On \(x\ge 0\) the graph passes the horizontal line test.
4. New range is \(y\ge 0\); the inverse is \[ f^{-1}(y)=\sqrt{y}\qquad(y\ge 0). \]

Example 2 – Sine function

1. Original: \(f(x)=\sin x\) on \(\mathbb{R}\). Not one‑to‑one.
2. Restrict domain to \(\displaystyle\left[-\frac{\pi}{2},\;\frac{\pi}{2}\right]\). This interval contains each \(y\in[-1,1]\) exactly once.
3. Inverse on this restricted domain is the arcsine function: \[ f^{-1}(y)=\arcsin y,\qquad -1\le y\le 1. \]

8. Sketching the inverse – reflection in the line \(y=x\)

The graph of an inverse function is the mirror image of the original graph in the line \(y=x\). To sketch:

1. Draw the line \(y=x\) on the same set of axes.
2. For any point \((a,b)\) on the graph of \(y=f(x)\), plot the reflected point \((b,a)\). Connecting all reflected points gives the graph of \(y=f^{-1}(x)\).
3. For a linear function \(f(x)=3x-7\) the reflection is the line \(y=\frac{x+7}{3}\), which is exactly the algebraic inverse found earlier.

9. Finding the inverse of a one‑to‑one function (step‑by‑step)

1. Start with the equation \(y=f(x)\).
2. Swap the symbols \(x\) and \(y\): \(x=f(y)\).
3. Solve this new equation for \(y\) in terms of \(x\).
4. Rename the solved variable \(y\) as \(x\); the resulting expression is \(f^{-1}(x)\).

Example: \(f(x)=e^{2x}\) (domain \(\mathbb{R}\), range \(>0\)).

1. \(y=e^{2x}\).
2. Swap: \(x=e^{2y}\).
3. Take natural logarithm: \(\ln x = 2y\) → \(y=\frac12\ln x\).
4. Rename: \(f^{-1}(x)=\dfrac12\ln x\) (valid for \(x>0\)).

10. Key points to remember

• A function has an inverse iff it is bijective (one‑to‑one + onto).
• The horizontal line test is a quick visual way to check the one‑to‑one property.
• If two different \(x\)-values give the same \(y\), the “inverse’’ would have to assign one \(y\) to several \(x\)-values – this is not allowed for a function.
• Restricting the domain (or range) can turn a many‑one function into a bijection, allowing a legitimate inverse to be defined.
• When writing an inverse, always start with \(y=f(x)\), interchange \(x\) and \(y\), solve for the new \(y\), then rename the variable.
• For composite functions remember that \((g\circ f)(x)=g(f(x))\) and the order cannot be interchanged in general.
• The graph of \(f^{-1}\) is the reflection of the graph of \(f\) in the line \(y=x\).
• Absolute‑value transformations are obtained by reflecting any part of the graph that lies below the \(x\)-axis upwards.

11. Sample exam question (IGCSE style)

Question: The function \(f\) is defined by \(f(x)=3x-7\) for all real numbers \(x\).

1. Explain, using appropriate terminology, why \(f\) has an inverse.
2. Find an expression for \(f^{-1}(x)\).

Answer outline:

1. Because the function is linear with a non‑zero slope, it is strictly increasing; therefore every horizontal line meets the graph exactly once. Hence \(f\) is one‑to‑one. The range of a linear function with real coefficients is all real numbers, so \(f\) is also onto. Consequently \(f\) is bijective and possesses an inverse.
2. Start with \(y=3x-7\). Solve for \(x\): \[ x=\frac{y+7}{3}. \] Rename the independent variable \(y\) as \(x\): \[ f^{-1}(x)=\frac{x+7}{3}. \]

12. Quick reference table