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Chapter 6: Problem 5

Plot a graph of the following functions. In each case state the domain and the range of the function. (a) \(f(x)=3 x+2,-2 \leq x \leq 5\) (b) \(g(x)=x^{2}+4,-2 \leq x \leq 3\) (c) \(p(t)=2 t^{2}+8,-2 \leq t \leq 4\) (d) \(f(t)=6-t^{2}, 1 \leq t \leq 5\)

Short Answer

Expert verified
Question: Determine the domain and range for each function, and briefly describe how the graph looks. (a) \(f(x)=3 x+2,-2 \leq x \leq 5\) (b) \(g(x)=x^{2}+4,-2 \leq x \leq 3\) (c) \(p(t)=2 t^{2}+8,-2 \leq t \leq 4\) (d) \(f(t)=6-t^{2}, 1 \leq t \leq 5\)

Step by step solution

01

Identify the Domain

The domain of the function is given explicitly, and it is \(-2 \leq x \leq 5\).
02

Identify the Range

The range is determined by what the function does to the input values. In this case, it is a linear function, and it increases in value for larger x values, so the range will be from the smallest output to the largest output. The smallest output occurs when \(x = -2\), resulting in \(f(-2) = 3(-2) + 2 = -4\). The largest output happens when \(x = 5\), giving \(f(5) = 3(5) + 2 = 17\). Thus, the range is \(-4 \leq f(x) \leq 17\).
03

Plot the Function

To plot the function, first find the points corresponding to the minimum and maximum x values: \((-2,-4)\) and \((5,17)\). Then, draw a line segment connecting these two points, representing the linear function. (b) \(g(x)=x^{2}+4,-2 \leq x \leq 3\)
04

Identify the Domain

The domain of the function is given explicitly, and it is \(-2 \leq x \leq 3\).
05

Identify the Range

This is a quadratic function, so the output values will vary quadratically with the input values. However, the minimum value of a quadratic function only occurs once. Since the coefficient of x^2 is positive, the graph will be U-shaped, with the minimum occurring at the vertex. Adding 4 shifts the graph vertically 4 units up so the minimum value will be when x is halfway between -2 and 3. This happens at \(x = \frac{-2+3}{2} = 0.5\), resulting in \(g(0.5) = (0.5)^2 + 4 = 4.25\). So, as x goes from \(-2\) to \(3\), the range goes from the minimum value of \(4.25\) to the maximum value of \(g(3)= (3)^2 + 4 = 13\). Thus, the range is \(4.25 \leq g(x) \leq 13\).
06

Plot the Function

To plot the function, first find the vertex and the corresponding points for minimum and maximum x values. The vertex is \((0.5,4.25)\). Connect minimum and maximum points wth a U-shaped curve. (c) \(p(t)=2 t^{2}+8,-2 \leq t \leq 4\)
07

Identify the Domain

The domain of the function is given explicitly, and it is \(-2 \leq t \leq 4\).
08

Identify the Range

This is also a quadratic function. The graph of a quadratic function has a parabolic shape. Since the coefficient of t^2 is positive, the graph opens upward. The minimum value occurs at the endpoint of the domain with the smallest value: \(p(-2) = 2(-2)^2 + 8 = 24\). The maximum value occurs at the endpoint with the largest value: \(p(4)= 2(4)^2 + 8 = 72\). Thus, the range is \(24 \leq p(t) \leq 72\).
09

Plot the Function

To plot the function, first plot the minimum and maximum points \((-2,24)\) and \((4,72)\). Then, connect these points with a U-shaped curve that represents the quadratic function. (d) \(f(t)=6-t^{2}, 1 \leq t \leq 5\)
10

Identify the Domain

The domain of the function is given explicitly, and it is \(1 \leq t \leq 5\).
11

Identify the Range

This quadratic function has a negative coefficient for the t^2 term, so the graph is inverted and opens downward. The maximum value will occur when t is halfway between 1 and 5: \(t = \frac{1+5}{2} = 3\). Thus, the maximum value is \(f(3) = 6 - (3)^2 = -3\). The minimum value occurs at either endpoint of the domain, whichever results in a smaller output. Since the graph opens downward, the minimum value occurs at the endpoint with the largest value: \(f(5) = 6 - (5)^2 = -19\). Thus, the range is \(-19 \leq f(t) \leq -3\).
12

Plot the Function

To plot the function, find the maximum and minimum points, which are \((3,-3)\) and \((5,-19)\). Then, connect these points with an upside-down U-shaped curve to represent the quadratic function.

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