Math Problem Statement

Use functions, f(x), g(x), h(x) and p(t) to answer the questions provided

The f(x) graph includes a line where point 1 is -5x axis -4y axis, it intersects at 0x axis, 0y axis

and the end of the line, point 2 is 5y axis and 4x axis.

underneath we have a table that shows

g(x): {(-10,36), (2,50), (19,78), (23,74), (30,2), (41,54),

(47,52), (48,73), (52,-2), (71,-9)}

also

t is 28,51,69,77,84,85,100,106,110,119

p(t) is 87,23,86,51,94,111,39,85,62,31

and below that we observe that h(x) = -18x + 20

Evaluate f(3) = what?

Determine x when f(x) = 1

evaluate g(52) = what?

evaluate h(12) = what?

determine x when h(x) = -142

evaluate p(85)

determine t when p(t) = 51

Solution

Let's go through each function step-by-step based on the provided information:

1. Evaluate f(3)f(3):

The graph of f(x)f(x) is described as a line that passes through two points:

  • Point 1: (5,4)(-5, -4)
  • Point 2: (5,4)(5, 4)

Using these two points, we can find the equation of the line f(x)f(x).

The slope of the line mm is calculated as:

m=y2y1x2x1=4(4)5(5)=4+45+5=810=0.8m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-4)}{5 - (-5)} = \frac{4 + 4}{5 + 5} = \frac{8}{10} = 0.8

Now that we have the slope, we can use the point-slope form of the line equation yy1=m(xx1)y - y_1 = m(x - x_1).

Using point 1 (5,4)(-5, -4):

y + 4 = 0.8(x + 5) \\ y = 0.8x + 4 - 4 \\ f(x) = 0.8x$$ Thus, the equation of the line is $$f(x) = 0.8x$$. Now, evaluate $$f(3)$$: $$f(3) = 0.8 \times 3 = 2.4$$ ### Answer: $$f(3) = 2.4$$ --- ### 2. Determine $$x$$ when $$f(x) = 1$$: We have the equation $$f(x) = 0.8x$$. Set this equal to 1 and solve for $$x$$: $$0.8x = 1 \\ x = \frac{1}{0.8} = 1.25$$ ### Answer: $$x = 1.25$$ when $$f(x) = 1$$ --- ### 3. Evaluate $$g(52)$$: The table for $$g(x)$$ includes the value $$g(52)$$. From the table, we observe: $$g(52) = -2$$ ### Answer: $$g(52) = -2$$ --- ### 4. Evaluate $$h(12)$$: The function $$h(x)$$ is given as $$h(x) = -18x + 20$$. Now, evaluate $$h(12)$$: $$h(12) = -18(12) + 20 = -216 + 20 = -196$$ ### Answer: $$h(12) = -196$$ --- ### 5. Determine $$x$$ when $$h(x) = -142$$: Set $$h(x) = -142$$ in the equation $$h(x) = -18x + 20$$ and solve for $$x$$: $$-18x + 20 = -142 \\ -18x = -142 - 20 \\ -18x = -162 \\ x = \frac{-162}{-18} = 9$$ ### Answer: $$x = 9$$ when $$h(x) = -142$$ --- ### 6. Evaluate $$p(85)$$: The table for $$p(t)$$ includes the value $$p(85)$$. From the table, we observe: $$p(85) = 111$$ ### Answer: $$p(85) = 111$$ --- ### 7. Determine $$t$$ when $$p(t) = 51$$: The table for $$p(t)$$ includes the values of $$t$$ and corresponding $$p(t)$$. From the table, we observe that $$p(t) = 51$$ when $$t = 77$$. ### Answer: $$t = 77$$ when $$p(t) = 51$$ --- Let me know if you'd like more details or if you have any further questions! Here are 5 related questions to expand on the topics we've discussed: 1. How would you calculate the inverse of the function $$f(x)$$? 2. If the slope of the line in $$f(x)$$ changed, how would that affect the graph? 3. How can you estimate the behavior of $$g(x)$$ between values not in the table? 4. What is the range of the function $$h(x) = -18x + 20$$? 5. If $$p(t)$$ were modeled as a polynomial, what degree would fit the data? **Tip:** Always verify points used to calculate the slope of a line for accuracy—small errors in the coordinates can lead to incorrect results in the equation.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Table of Values
Algebraic Functions

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Point-slope form: y - y1 = m(x - x1)
Linear function: f(x) = mx + b
Linear equation for h(x): h(x) = -18x + 20

Theorems

Slope of a line
Linear equation solution methods

Suitable Grade Level

Grades 9-11