Problem 52
Question
the domain of each piecewise function is \((-\infty, \infty)\) a. Graph each function. b. Use your graph to determine the function's range. $$ f(x)=\left\\{\begin{array}{rll} -\frac{1}{2} x^{2} & \text { if } & x<1 \\ 2 x+1 & \text { if } & x \geq 1 \end{array}\right. $$
Step-by-Step Solution
Verified Answer
The range of the given piecewise function is \( [-0.5, \infty) \).
1Step 1: Graph the first part of the function
We plot the function \( -\frac{1}{2}x^{2} \) for all \( x < 1 \). This will result in a downward facing parabola.
2Step 2: Graph the second part of the function
Next, we plot the function \( 2x+1 \) for all \( x \geq 1 \). This will result in an upward sloping line starting from the point (1,3).
3Step 3: Join the two pieces
Join the two parts of the graph at \( x = 1 \). Since the second part is defined for \( x \geq 1 \), the point (1,3) should be filled indicating that this point is included in the function's domain.
4Step 4: Determine the range
The range of this piecewise function is all the y-values that the function reaches. By observing the graph, we can see that the function reaches every y-value greater than or equal to -0.5. So the range of the function is \( [-0.5, \infty) \).
Key Concepts
Graphing Piecewise FunctionsDomain and Range of FunctionsQuadratic Functions
Graphing Piecewise Functions
When it comes to graphing piecewise functions, it can seem like a challenging task at first, but it's actually just like putting together pieces of a puzzle. A piecewise function is a function that has different expressions based on different intervals of the input value, or the 'x' value.
For example, in the provided exercise, we have a function that changes its formula at the point where x equals 1. To graph this, you'll need to consider each part of the function separately. The first segment, given by a quadratic function \( -\frac{1}{2}x^2 \) for \( x < 1 \), bends downwards and will look like a half of a parabola.
The second segment starts at \( x = 1 \) and continues onward, defined by the linear function \( 2x + 1 \). This segment will graph as a straight line that goes upwards from the point \( (1,3) \). When these two parts are graphed on the same set of axes, you carefully connect them at their boundary without overlapping.
Remember to pay attention to whether the pieces include the boundary point. If the function value at the boundary is defined with an 'equals' sign (\( \geq 1 \)), you'll fill in that boundary point on the graph; otherwise, it's left as an open circle. It's like knitting together the fabric of the graph, where each piece neatly attaches to the next.
For example, in the provided exercise, we have a function that changes its formula at the point where x equals 1. To graph this, you'll need to consider each part of the function separately. The first segment, given by a quadratic function \( -\frac{1}{2}x^2 \) for \( x < 1 \), bends downwards and will look like a half of a parabola.
The second segment starts at \( x = 1 \) and continues onward, defined by the linear function \( 2x + 1 \). This segment will graph as a straight line that goes upwards from the point \( (1,3) \). When these two parts are graphed on the same set of axes, you carefully connect them at their boundary without overlapping.
Remember to pay attention to whether the pieces include the boundary point. If the function value at the boundary is defined with an 'equals' sign (\( \geq 1 \)), you'll fill in that boundary point on the graph; otherwise, it's left as an open circle. It's like knitting together the fabric of the graph, where each piece neatly attaches to the next.
Domain and Range of Functions
Understanding the domain and range is crucial because it tells us what the function is physically capable of handling and what it can produce. The domain is the set of all possible input values (x-values), whereas the range is the set of all possible output values (y-values).
In our example, the domain is explicitly stated as \( (-\infty, \infty) \), which means that there is no restriction on the x-values; the function can take any value from negative to positive infinity. However, determining the range is more about examining what y-values are achieved when x takes on every possible value.
After graphing the piecewise function, you can visually inspect where the y-values lie. For the downward parabola, it starts from positive infinity and decreases to its lowest point (vertex), and for the linear piece, it goes from the y-value at the boundary (\( x=1 \) in this case) to positive infinity. The lowest point the parabola reaches corresponds to the beginning of the range, and since it's a continuous curve, all y-values from this point are included up to infinity, giving us the range as \( [-0.5, \infty) \).
In our example, the domain is explicitly stated as \( (-\infty, \infty) \), which means that there is no restriction on the x-values; the function can take any value from negative to positive infinity. However, determining the range is more about examining what y-values are achieved when x takes on every possible value.
After graphing the piecewise function, you can visually inspect where the y-values lie. For the downward parabola, it starts from positive infinity and decreases to its lowest point (vertex), and for the linear piece, it goes from the y-value at the boundary (\( x=1 \) in this case) to positive infinity. The lowest point the parabola reaches corresponds to the beginning of the range, and since it's a continuous curve, all y-values from this point are included up to infinity, giving us the range as \( [-0.5, \infty) \).
Quadratic Functions
Quadratic functions describe parabolas, which are symmetrical, u-shaped curves that can open upward or downward. In general form, a quadratic function is written as \( f(x) = ax^2 + bx + c \), where 'a' determines the direction of the opening and the 'width' of the parabola, 'b' affects the position of its axis of symmetry, and 'c' is the y-intercept.
In the context of the exercise, for \( x < 1 \) we have a quadratic equation where \( a = -\frac{1}{2} \) which means our parabola opens downwards. There's no 'b' term, so the axis of symmetry is the y-axis, and since there's no 'c' term either, the vertex is at the origin (0,0).
Understanding the characteristics of quadratic functions is key to not only graphing them but also predicting the behavior of real-world scenarios they represent, such as the trajectory of a ball thrown in the air. The properties of their symmetry and their maxima or minima (depending on the direction they open) can tell us a lot about the solutions to the problems they're applied to.
In the context of the exercise, for \( x < 1 \) we have a quadratic equation where \( a = -\frac{1}{2} \) which means our parabola opens downwards. There's no 'b' term, so the axis of symmetry is the y-axis, and since there's no 'c' term either, the vertex is at the origin (0,0).
Understanding the characteristics of quadratic functions is key to not only graphing them but also predicting the behavior of real-world scenarios they represent, such as the trajectory of a ball thrown in the air. The properties of their symmetry and their maxima or minima (depending on the direction they open) can tell us a lot about the solutions to the problems they're applied to.
Other exercises in this chapter
Problem 52
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Find a. \((f \circ g)(x) \quad \) b. \((g \circ f)(x) \quad \) c. \((f \circ g)(2) \quad \) d. \((g \circ f)(2)\) $$f(x)=3 x, g(x)=x-5$$
View solution Problem 52
Graph each equation in a rectangular coordinate system. $$x=5$$
View solution