Skill: Identify horizontal translations of functions

A horizontal translation changes the input of a function, so the graph moves left or right while its output values stay the same.

In the form

$$y=f(x-h),$$

• if $h>0$, the graph shifts $h$ units right;
• if $h<0$, the graph shifts $\vert h \vert$ units left.

The sign can feel backward: an expression such as $x-3$ means a shift right 3, not left 3.

Example: Let

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

and define

$$g(x)=f(x-3).$$

Describe the translation, point mapping, domain, and range of $g$.

Step 1: Identify the value of $h$.

Compare $g(x)=f(x-3)$ with $f(x-h)$. Here,

$$h=3.$$

Since $h$ is positive, the graph shifts 3 units right.

Step 2: Apply the point mapping.

A point $(x,y)$ on the original graph moves according to

$$(x,y)\mapsto(x+h,y).$$

Therefore,

$$(x,y)\mapsto(x+3,y).$$

For example, the point $(0,0)$ on $y=\sqrt{x}$ moves to $(3,0)$ on $y=g(x)$.

Step 3: Write the new function and domain.

Because

$$g(x)=f(x-3)=\sqrt{x-3},$$

the expression inside the square root must be nonnegative:

$$x-3\ge 0,$$

so

$$x\ge 3.$$

The domain is

$$[3,\infty).$$

Step 4: Identify the range.

A horizontal translation does not change the output values. The original function $f(x)=\sqrt{x}$ has range

$$[0,\infty),$$

so $g$ has the same range:

$$[0,\infty).$$

Thus, $g(x)=f(x-3)$ is the graph of $f$ shifted 3 units right. Its domain shifts from $[0,\infty)$ to $[3,\infty)$, while its range remains $[0,\infty)$.