When does an inverse exist?

Restrict or extend the domain without changing the rule

A restriction keeps the same rule and uses a subset of the original domain. To make an inverse possible, keep inputs for which no two different inputs share an output. On a parabola, retaining one side of the vertex achieves this.

Keep the endpoint condition

The source compares $f(x)=(x+2)^2$ on $\mathbb R$ with $g(x)=(x+2)^2$ on $x>-2$. The full parabola has vertex $(-2,0)$ and y-intercept 4; the restricted graph has an open point at the vertex and only the increasing right branch. The range of the restricted function is $(0,\infty)$: the vertex output 0 is approached but never reached.

Thus $D_g\subseteq D_f$. These are different functions because their domains differ. The source’s “Extensions” note reverses the comparison: the same rule with a larger domain extends the smaller-domain function. Restriction can recover one-to-one behaviour; extension may lose it.

Example 5 retains the Teacher’s choice $[0,3]$ for the printed “largest possible domain” request. The solution explains the ambiguity: it is the longest injective interval, while $[-2,0]$ is another inclusion-maximal interval. The original wording is preserved.

An inverse reverses each permitted mapping

If f maps x to y, its inverse maps y back to x. The source’s arrow diagram labels the original domain as the inverse range and the original range as the inverse domain. For $f(x)=2x+1$, $x>0$, solving $f(x)=11$ gives $2x+1=11$, then $x=5$: equivalently $f^{-1}(11)=5$. A one-to-one function and its inverse express the same relationship with the input/output roles reversed—the source’s “Equivalence” connection.

Find the inverse in four steps

1. Check that the function is one-to-one on its stated domain.
2. Set $y=f(x)$ and solve for the original input $x$ in terms of $y$.
3. Use the original domain to choose the sign or branch. Do not keep both square roots when only one is allowed.
4. Write $f^{-1}(y)$, then rename its input to $x$. State the inverse domain and range, including endpoint conditions.

The inverse exists as a function on the original range when f is one-to-one. In particular,

$D_{f^{-1}}=R_f,\qquad R_{f^{-1}}=D_f.$

Define an inverse with both its rule and domain. Example 6 supplies the linear rearrangement; Example 7 rejects the positive root because its original inputs are negative, preserving the open endpoints.

Reflection swaps coordinates and asymptotes

Use the same scale on both axes. Reflect $(a,b)$ to $(b,a)$ in $y=x$; draw that reference line even when it is not explicitly requested. A horizontal asymptote $y=k$ becomes a vertical asymptote $x=k$, and conversely. Example 8 keeps the complete Teacher exponential/logarithm sketch, with intercepts $(0,1),(1,0)$ and asymptotes $y=0,x=0$.

For $a>0$, $a\ne1$, taking logarithms to base a in $y=a^x$ gives $x=\log_a y$. Exponentials and logarithms with the same base are inverses. The source illustrates bases2 and e, including $(1,e)\leftrightarrow(e,1)$. Its statement that “any logarithmic function is strictly increasing” needs the qualification $a>1$. For $0<a<1$, the logarithm is strictly decreasing; both cases have domain $(0,\infty)$ and vertical asymptote $x=0$.

Principal inverse trigonometric branches

Unrestricted sine, cosine and tangent are not one-to-one. Restrict before reversing:

Sine

Restrict $\sin x$ to $[-\pi/2,\pi/2]$.

• Inverse: $\sin^{-1}x$.
• Domain: $[-1,1]$.
• Range: $[-\pi/2,\pi/2]$.

Cosine

Restrict $\cos x$ to $[0,\pi]$.

• Inverse: $\cos^{-1}x$.
• Domain: $[-1,1]$.
• Range: $[0,\pi]$.

Tangent

Restrict $\tan x$ to $(-\pi/2,\pi/2)$.

• Inverse: $\tan^{-1}x$.
• Domain: $\mathbb R$.
• Range: $(-\pi/2,\pi/2)$.

The sine and cosine endpoints are included. Tangent is undefined at the restriction’s endpoints, and arctangent approaches its range boundaries without reaching them.

The figures retain the repeated-output sine example, the closed endpoints of the sine/cosine restrictions and their inverses, and the vertical tangent/horizontal arctangent asymptotes at $\pm\pi/2$. In this context the superscript −1 denotes an inverse function, not a reciprocal. Angles use radians. The source’s principal branches are standard choices; other injective restrictions define different inverse branches.

One counterexample is enough to disprove one-to-one behaviour. To prove it, cover the entire allowed domain; strict increase or strict decrease is a useful sufficient condition. Restricting a domain keeps the rule but defines a different function. Always give both the inverse rule and its domain, preserving open and closed endpoints.