Concept explainers
In Exercises 1-4, graph the given quadratic function. Give each function’s domain and range.
1. f(x) = (x − 3)2 − 4
To sketch: The graph of the function and then find the domain and range of the function
Answer to Problem 1MC
The domain of the function is
The range of the function is
Explanation of Solution
Definition used:
The domain of a function is a set of real values of the independent variable such that the corresponding dependent values also will be real.
The range of a function is defined as a set of all possible functional values, which is denoted by
Calculation:
Consider the quadratic function
It is known that the standard form of the quadratic equation is
Compare the given quadratic equation
Thus, the vertex of the parabola is
Find the x intercept by substituting 0 for
The x intercepts are 1 and 5.
That is, the parabola passes through
Find the y intercept by substituting 0 for x in
The y intercept is 5.
That is, the parabola passes through
Use the above information to draw the graph of the function
From Figure 1, it is observed that the parabola is open upward and the vertex of the parabola is
For all the real values of x, the function
Therefore, there is no restriction for the domain of the function.
Hence, the domain of the function is
Note that, the vertex of the parabola is
Since y coordinate of the vertex is −4, the functional values will fall on or above −4.
Hence, the range of the function is
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Chapter 3 Solutions
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