Examples of range in the following topics:
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- Example 1: Determine the domain and range of each graph pictured below:
- The range for the graph f(x)=−121x3, is R.
- Example 2:
Determine the domain and range of each graph pictured below:
- The range of the blue graph is all real numbers, R.
- Use the graph of a function to determine its domain and range
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- The domain of a function is the set of all possible input values that produce some output value range
- A function is the relation that takes the inputs of the domain and output the values in the range.
- As you can see in the illustration, each value of the domain has a green arrow to exactly one value of the range.
- The oval on the left is the domain of the function f, and the oval on the right is the range.
- The green arrows show how each member of the domain is mapped to a particular value of the range.
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- Note that the domain of the inverse function is
the range of the original function, and vice versa.
- Note the domain and range of each function.
- $\displaystyle{y = \cos^{-1}x \quad \text{has domain} \quad \left[-1, 1\right] \quad \text{and range} \quad \left[0, \pi\right]}$
- To find the domain and range
of inverse trigonometric functions, we switch the domain and range of the
original functions.
- Describe the characteristics of the graphs of the inverse trigonometric functions, noting their domain and range restrictions
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- In other words, every element of the function's range corresponds to exactly one element of its domain.
- Another way to determine if the function is one-to-one is to make a table of values and check to see if every element of the range corresponds to exactly one element of the domain.
- The ordered pairs (−2,4) and (2,4) do not pass the definition of one-to-one because the element 4 of the range corresponds to to −2 and 2.
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- Example 2: Graph the function and determine its domain and range:
- The range begins at the lowest y-value, y=0 and is continuous through positive infinity.
- Therefore the range of the piecewise function is also the set of all real numbers greater than or equal to 0, or all non-negative values: y≥0.
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- If both are solved for x, we will see the full range of possible values of x.
- We now have two ranges of solutions to the absolute value inequality; x>4 and x<−4.
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- The range of the function is all real numbers.
- The range of the square root function is all non-negative real numbers, whereas the range of the logarithmic function is all real numbers.
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- Firstly, doing so allows one to plot a very large range of data without losing the shape of the graph.
- Secondly, it allows one to interpolate at any point on the plot, regardless of the range of the graph.
- A key point about using logarithmic graphs to solve problems is that they expand scales to the point at which large ranges of data make more sense.
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- If f is an invertible function with domain X and range Y, then
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- Making algebraic computations with variables as if they were explicit numbers allows one to solve a range of problems in a single computation.