The range of a function is the set of all possible output values (y-values) that the function can produce based on its formula or rule.
How to Find the Range of a Function
- Using the formula:
- Write the function in the form y=f(x)y=f(x)y=f(x).
- Solve the equation for xxx in terms of yyy to get x=g(y)x=g(y)x=g(y).
- The domain of g(y)g(y)g(y) (values for which xxx is defined) will be the range of f(x)f(x)f(x).
- For example, for a rational function f(x)=14x−3f(x)=\frac{1}{4x-3}f(x)=4x−31, solving for xxx in terms of yyy excludes values that make the denominator zero, which defines the range.
- Using a graph:
- Sketch or look at the graph of the function.
- Identify the vertical spread of the graph (the y-values covered).
- The range is all y-values that the graph touches or covers.
- Using ordered pairs (relation):
- List all y-values from the given relation (set of ordered pairs (x,y)(x,y)(x,y)).
- The range is the set of all these y-values.
Common Rules for Popular Functions
- Linear functions have a range of all real numbers.
- Quadratic functions y=a(x−h)2+ky=a(x-h)^2+ky=a(x−h)2+k have ranges depending on aaa:
- y≥ky\geq ky≥k if a>0a>0a>0
- y≤ky\leq ky≤k if a<0a<0a<0
- Square root functions have range y≥0y\geq 0y≥0.
- Exponential functions have range y>0y>0y>0.
- Logarithmic functions have range of all real numbers.
In summary, the range represents all possible output values for yyy that come from substituting all allowed xxx values into the function and can be found algebraically, graphically, or from the function's relation.
