Function notation to write g in terms of f is a way of describing one function using another function as the starting point. But instead of creating a completely new formula from scratch, you take a known function, such as f(x), and apply changes to it, such as shifts, stretches, reflections, or substitutions. This idea is especially useful in algebra, graphing, transformations, and later mathematics such as calculus, physics, economics, and computer science.
Quick note before moving on.
Introduction to Function Notation
Function notation is a compact way to show how an input value is connected to an output value. The expression
[ f(x) ]
means “the value of the function f when the input is x.” To give you an idea, if
[ f(x)=2x+3, ]
then
[ f(4)=2(4)+3=11. ]
When we write g in terms of f, we express the function g using f. This often means that g(x) is built from f(x) by changing the input, the output, or both.
To give you an idea, if
[ f(x)=x^2, ]
then we might define
[ g(x)=f(x)+5. ]
Since (f(x)=x^2), this means
[ g(x)=x^2+5. ]
So, g is the function f shifted upward by 5 units And that's really what it comes down to. Practical, not theoretical..
What Does “Write g in Terms of f” Mean?
To write g in terms of f means to express g(x) using the function f. The goal is to show how g depends on f The details matter here..
For example:
[ g(x)=f(x)+7 ]
says that g(x) is always 7 more than f(x).
Another example:
[ g(x)=f(x-3) ]
says that g(x) uses the value of f three units earlier than (x) That's the part that actually makes a difference. Nothing fancy..
A third example:
[ g(x)=2f(x) ]
says that g(x) is twice the output of f(x).
In each case, g is not completely independent of f. It is defined by modifying f.
Basic Forms of g in Terms of f
There are several common ways to write g in terms of f.
1. Vertical Shifts
A vertical shift changes the output values of a function.
If
[ g(x)=f(x)+k, ]
then the graph of g is the graph of f shifted vertically by (k) units.
- If (k>0), the graph moves up.
- If (k<0), the graph moves down.
Example:
If
[ f(x)=x^2 ]
and
[ g(x)=f(x)+4, ]
then
[ g(x)=x^2+4. ]
The graph of (g) is the graph of (f) shifted up 4 units.
2. Horizontal Shifts
A horizontal shift changes the input values of a function.
If
[ g(x)=f(x-h), ]
then the graph of g is the graph of f shifted horizontally by (h) units.
- If (h>0), the graph moves right.
- If (h<0), the graph moves left.
This can confuse students because the sign inside the function often appears to do the opposite of what one might expect.
Example:
If
[ f(x)=x^2 ]
and
[ g(x)=f(x-2), ]
then
[ g(x)=(x-2)^2. ]
The graph of g is the graph of f shifted 2 units to the right.
3. Vertical Stretches and Compressions
A vertical stretch or compression changes the output values by multiplying them.
If
[ g(x)=a f(x), ]
then:
- If (|a|>1), the graph becomes vertically stretched.
- If (0<|a|<1), the graph becomes vertically compressed.
- If (a<0), the graph is reflected across the x-axis.
Example:
If
[ f(x)=x^2 ]
and
[ g(x)=3f(x), ]
then
[ g(x)=3x^2. ]
The graph of g is stretched vertically by a factor of 3.
4. Horizontal Stretches and Compressions
A horizontal stretch or compression changes the input values inside the function.
If
[ g(x)=f(bx), ]
then:
- If (|b|>1), the graph is horizontally compressed.
- If (0<|b|<1), the graph is horizontally stretched.
