STEP-BY-STEP LESSON · §11.1
Real-Valued Functions of a Real Variable and Their Graphs
Read a function graph as a relationship between input and output.
Before you begin
Coordinates of a function graph
The point (x,y) records a horizontal input x and vertical output y. For the graph of f, y=f(x). Δx denotes a change in input, not a new independent axis.
Function, domain and codomain
A function assigns exactly one permitted output to every input in its domain. Its codomain is the declared output set; some members of that set may never be reached. The same formula on another domain or codomain can have different properties. f(x) names the output at x; it is not multiplication.
Symbols
- y=f(x)
- output y at input x
- ℝ
- The set of real numbers
- Δx
- Change in input: new input minus old input
Definitions and notation for this topic
Intervals and endpoints
[a,b]; (a,b); [0,∞)
For real a<b, [a,b] includes both endpoints; (a,b) excludes both. A square bracket includes its finite endpoint and a round bracket excludes it.
Infinity is not a real endpoint to include, so ∞ always has a round bracket in this notation. Intervals contain real values, not only integers.
Separate glossary example
[0,∞)={x∈ℝ:x≥0}; (−1,2]={x∈ℝ:−1<x≤2}.
Function evaluation
f(x)
For f:A→B and x∈A, f(x) is the unique output in B assigned to input x. It is an element of B, not generally a set.
The declared function determines the operation; f(x) does not mean f multiplied by x. Distinguish an input element x from an input subset S in f(S).
Separate glossary example
For f:ℤ→ℤ, f(x)=x², f(−3)=9; the image of the set {−3} is f({−3})={9}.
Change in an input or output
Δx=x_new−x_old; Δy=f(x_new)−f(x_old)
Δ denotes a finite change: the new value minus the old value. For a graph y=f(x), specify both inputs before comparing their outputs.
The output change depends on the function and starting point. Δx is one quantity, not multiplication by an independently defined variable Δ.
Separate glossary example
For f(x)=x², moving from x=3 to x=4 gives Δx=1 and Δy=16−9=7.
Step by step
Step 1 / 6
Specify the domain
First determine which inputs are allowed. The graph consists of points (x,f(x)) for those inputs.
The same formula on different domains can have different properties.
Worked example
Is f(x)=x² increasing on all ℝ? What about x≥0?
- On ℝ, −2<−1 gives 4>1, so the function is not increasing on all ℝ.
- For 0≤x₁<x₂, the difference x₂²−x₁²=(x₂−x₁)(x₂+x₁)>0.
- Thus the function is strictly increasing on nonnegative inputs.
Optional self-check
For f(x)=−2x+1, x rises from 1 to 4. What happens to f(x), and what does it say about the slope?
Show answer
Outputs change from −1 to −7, a decrease of 6. The input change is 3, multiplied by the negative slope −2.