Question

use the graph of g to solve exercises 71 - 76.

1. find g(-4).
2. find g(1).
3. find g(-10).
4. find g(10).
5. for what value of x is g(x) = 1?
6. for what value of x is g(x) = -1?

in exercises 77 - 92, use the graph to determine a. the functions domain; b. the functions range; c. the x - intercepts, if any; d. the y - intercept, if any; and e. the missing function values, indicated by question marks, below each graph.
77.
graph of y = f(x) with f(-2) =? and f(2) =?
78.
graph of y = f(x) with f(-2) =? and f(2) =?
79.
graph of y = f(x) with f(-1) =? and f(3) =?
80.
graph of y = f(x) with f(-4) =? and f(3) =?
81.
graph of y = f(x) with f(3) =?
82.
graph of y = f(x) with f(-5) =?
83.
graph of y = f(x) with f(4) =?
84.
graph of y = f(x) with f(3) =?
85.
graph of y = f(x) with f(-1) =?
86.
graph of y = f(x) with f(-1) =?
87.
graph of y = f(x) with f(-4) =? and f(4) =?
88.
graph of y = f(x) with f(-4) =? and f(4) =?
89.
graph of y = f(x) with f(4) =?
90.
graph of y = f(x) with related description
91.
graph of y = f(x) with f(-5) + f(3) =?

Answer:

To solve these problems, we analyze the graphs of the functions to determine the required values. Let's take a few examples:

Problem 71: Find \( g(-4) \)

• Step 1: Locate \( x = -4 \) on the x - axis of the graph of \( y = g(x) \).
• Step 2: Find the corresponding \( y \) - value (function value) at \( x=-4 \). From the graph, when \( x = - 4 \), the \( y \) - value ( \( g(-4) \)) is 2.

Problem 75: For what value of \( x \) is \( g(x)=1 \)

• Step 1: Locate \( y = 1 \) on the y - axis.
• Step 2: Find the \( x \) - value(s) for which the function \( g(x) \) has a \( y \) - value of 1. From the graph, when \( g(x)=1 \), \( x = 0 \).

Problem 77: Find \( f(-2) \) and \( f(2) \)

• For \( f(-2) \):
• Step 1: Locate \( x=-2 \) on the x - axis of the graph of \( y = f(x) \).
• Step 2: The corresponding \( y \) - value at \( x = - 2 \) is - 4. So \( f(-2)=-4 \).
• For \( f(2) \):
• Step 1: Locate \( x = 2 \) on the x - axis.
• Step 2: The corresponding \( y \) - value at \( x = 2 \) is 0. So \( f(2)=0 \).

Problem 78: Find \( f(-2) \) and \( f(2) \)

• For \( f(-2) \):
• Step 1: Locate \( x=-2 \) on the x - axis of the graph of \( y = f(x) \) (a parabola - shaped graph).
• Step 2: The corresponding \( y \) - value at \( x=-2 \) is 4. So \( f(-2) = 4\).
• For \( f(2) \):
• Step 1: Locate \( x = 2 \) on the x - axis.
• Step 2: The corresponding \( y \) - value at \( x = 2 \) is - 4. So \( f(2)=-4 \).

Problem 79: Find \( f(-1) \) and \( f(3) \)

• For \( f(-1) \):
• Step 1: Locate \( x=-1 \) on the x - axis of the graph of \( y = f(x) \) (a V - shaped graph).
• Step 2: The corresponding \( y \) - value at \( x=-1 \) is 1. So \( f(-1)=1 \).
• For \( f(3) \):
• Step 1: Locate \( x = 3 \) on the x - axis.
• Step 2: The corresponding \( y \) - value at \( x = 3 \) is 3. So \( f(3)=3 \).

Problem 80: Find \( f(-4) \) and \( f(3) \)

• For \( f(-4) \):
• Step 1: Locate \( x=-4 \) on the x - axis of the graph of \( y = f(x) \) (a V - shaped graph).
• Step 2: The corresponding \( y \) - value at \( x=-4 \) is 3. So \( f(-4)=3 \).
• For \( f(3) \):
• Step 1: Locate \( x = 3 \) on the x - axis.
• Step 2: The corresponding \( y \) - value at \( x = 3 \) is 3. So \( f(3)=3 \).

Problem 81: Find \( f(3) \)

• Step 1: Locate \( x = 3 \) on the x - axis of the graph of \( y = f(x) \) (a curve - shaped graph).
• Step 2: The corresponding \( y \) - value at \( x = 3 \) is 2. So \( f(3)=2 \).

Problem 82: Find \( f(-5) \)

• Step 1: Locate \( x=-5 \) on the x - axis of the graph of \( y = f(x) \) (a curve - shaped graph).
• Step 2: The corresponding \( y \) - value at \( x=-5 \) is 3. So \( f(-5)=3 \).

Problem 83: Find \( f(4) \)

• Step 1: Locate \( x = 4 \) on the x - axis of the graph of \( y = f(x) \) (a curve - shaped graph).
• Step 2: The corresponding \( y \) - value at \( x = 4 \) is 3. So \( f(4)=3 \).

Problem 84: Find \( f(3) \)

• Step 1: Locate \( x = 3 \) on the x - axis of the graph of \( y = f(x) \) (a curve - shaped graph).
• Step 2: The corresponding \( y \) - value at \( x = 3 \) is 2. So \( f(3)=2 \).

Problem 86: Find \( f(-1) \)

• Step 1: Locate \( x=-1 \) on the x - axis of the graph of \( y = f(x) \) (a line - shaped graph).
• Step 2: The equation of the line can be found. The line has a slope \( m=\frac{y_2 - y_1}{x_2 - x_1}\). Taking two points, say \((0,5)\) and \((5,0)\), \(m=\frac{0 - 5}{5-0}=- 1\). The equation is \(y=-x + 5\). When \(x=-1\), \(y=-(-1)+5=6\)? Wait, no, from the graph, when \(x=-1\), the \(y\) - value is 6? Wait, looking at the graph, when \(x=-1\), the \(y\) - coordinate is 6? Wait, the graph of \(y = f(x)\) in problem 86: when \(x=-1\), moving up from \(x=-1\) on the x - axis, the \(y\) - value is 6? Wait, no, let's re - examine. The line goes from \((0,5)\) to \((5,0)\). The equation is \(y=-x + 5\). When \(x=-1\), \(y=-(-1)+5 = 6\). So \(f(-1)=6\)? Wait, no, maybe I made a mistake. Wait, the graph in problem 86: when \(x=-1\), the \(y\) - value is 6? Let's check the grid. If the x - axis is from - 2 to 5 and y - axis from - 3 to 6. At \(x=-1\), the point on the line is at \(y = 6\)? Wait, maybe the slope is \(-1\) and y - intercept 5. So \(f(-1)=-(-1)+5=6\).

Problem 87: Find \( f(-4) \) and \( f(4) \)

• For \( f(-4) \):
• Step 1: Locate \( x=-4 \) on the x - axis of the graph of \( y = f(x) \) (a piece - wise linear graph).
• Step 2: The corresponding \( y \) - value at \( x=-4 \) is - 5. So \( f(-4)=-5 \).
• For \( f(4) \):
• Step 1: Locate \( x = 4 \) on the x - axis.
• Step 2: The corresponding \( y \) - value at \( x = 4 \) is - 2. So \( f(4)=-2 \).

Final Answers (for some selected problems):

• \( g(-4)=\boldsymbol{2} \)
• \( g(x) = 1\) when \(x=\boldsymbol{0}\)
• \( f(-2)=\boldsymbol{-4}\), \(f(2)=\boldsymbol{0}\) (for problem 77)
• \( f(-2)=\boldsymbol{4}\), \(f(2)=\boldsymbol{-4}\) (for problem 78)
• \( f(-1)=\boldsymbol{1}\), \(f(3)=\boldsymbol{3}\) (for problem 79)
• \( f(-4)=\boldsymbol{3}\), \(f(3)=\boldsymbol{3}\) (for problem 80)
• \( f(3)=\boldsymbol{2}\) (for problem 81)
• \( f(-5)=\boldsymbol{3}\) (for problem 82)
• \( f(4)=\boldsymbol{3}\) (for problem 83)
• \( f(3)=\boldsymbol{2}\) (for problem 84)
• \( f(-1)=\boldsymbol{6}\) (for problem 86)
• \( f(-4)=\boldsymbol{-5}\), \(f(4)=\boldsymbol{-2}\) (for problem 87)