Hey Steve, let's break down your questions step by step with concrete examples to clear up any confusion!

First, let's recap the three graph transformation types you mentioned:

• Vertical/Horizontal Shift
• Vertical/Horizontal Stretch/Compression
• Reflection

Which transformations affect a function's domain and range? #

Your initial thought that stretches/compressions don't impact domain or range isn't quite right—let's go through each category in detail:

Vertical/Horizontal Shifts

• Horizontal shifts (e.g., transforming ( f(x) ) to ( f(x - a) )): Only alter the domain. For example, take ( f(x) = \sqrt{x} ), which has a domain of ( [0, +\infty) ) and range ( [0, +\infty) ). Shifting it right by 2 gives ( f(x-2) = \sqrt{x-2} ), whose domain becomes ( [2, +\infty) ) while the range stays unchanged at ( [0, +\infty) ).
• Vertical shifts (e.g., ( f(x) ) to ( f(x) + b )): Only alter the range. Using ( f(x) = x^2 ) (domain ( \mathbb{R} ), range ( [0, +\infty) )), shifting it up by 3 gives ( x^2 + 3 ), whose range becomes ( [3, +\infty) ) while the domain remains all real numbers.

Vertical/Horizontal Stretch/Compression

• Horizontal stretches/compressions (e.g., ( f(x) ) to ( f(kx) ), ( k \neq 0 )): Will change the domain when the original function's domain isn't all real numbers. For example, take ( f(x) = \sqrt{4 - x^2} ), which has a domain of ( [-2, 2] ). Compressing it horizontally by a factor of 2 gives ( f(2x) = \sqrt{4 - (2x)^2} = 2\sqrt{1 - x^2} ), whose domain shrinks to ( [-1, 1] )—the range stays ( [0, 2] ) here.
• Vertical stretches/compressions (e.g., ( f(x) ) to ( kf(x) ), ( k \neq 0 )): Will change the range when the original function's range isn't all real numbers. Using ( f(x) = \sin x ) (range ( [-1, 1] )), stretching it vertically by a factor of 3 gives ( 3\sin x ), whose range expands to ( [-3, 3] ) while the domain remains all real numbers.

Reflections

• Reflections over the y-axis (e.g., ( f(x) ) to ( f(-x) )): Only change the domain if the original domain isn't symmetric about the origin. For example, ( f(x) = \sqrt{x} ) (domain ( [0, +\infty) )) becomes ( f(-x) = \sqrt{-x} ), whose domain is ( (-\infty, 0] ); the range stays ( [0, +\infty) ).
• Reflections over the x-axis (e.g., ( f(x) ) to ( -f(x) )): Only change the range. Using ( f(x) = \sqrt{x} ) again, ( -f(x) = -\sqrt{x} ) has a range of ( (-\infty, 0] ) while the domain remains ( [0, +\infty) ).
• Reflections over the origin (a common type of reflection that flips over both axes): This single transformation changes both domain and range. For example, ( f(x) = \sqrt{x} ) (domain ( [0, +\infty) ), range ( [0, +\infty) )) becomes ( -f(-x) = -\sqrt{-x} ), whose domain is ( (-\infty, 0] ) and range is ( (-\infty, 0] )—both are completely altered.

Is there a single transformation that affects both domain and range? #

Absolutely! The origin reflection I mentioned above is a perfect example. Let's take another concrete case to drive this home:

• Start with ( f(x) = x + 1 ) where ( x \in [0, 2] ) (so domain ( [0, 2] ), range ( [1, 3] )).
• Reflect it over the origin: the transformed function is ( -f(-x) = -(-x + 1) = x - 1 ), and its domain becomes ( [-2, 0] ) (since we're replacing ( x ) with ( -x ), the original ( x \in [0,2] ) translates to ( -x \in [-2,0] )), while the range becomes ( [-3, -1] ). Both domain and range are entirely changed by this single reflection.

To wrap up your initial question: your hunch that shifts and reflections impact domain/range is correct, but stretches/compressions do affect either domain or range depending on whether they're horizontal or vertical—they aren't "neutral" across the board.

备注：内容来源于stack exchange，提问作者Steve