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

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

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

The graph of \( f(x) \) is described as a line that passes through two points:
- Point 1: \( (-5, -4) \)
- Point 2: \( (5, 4) \)

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

The slope of the line \( m \) is calculated as:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-4)}{5 - (-5)} = \frac{4 + 4}{5 + 5} = \frac{8}{10} = 0.8
\]

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

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

\[
y - (-4) = 0.8(x - (-5)) \\
y + 4 = 0.8(x + 5) \\
y = 0.8x + 4 - 4 \\
f(x) = 0.8x
\]

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.

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 and 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.

This detailed problem involves solving linear functions such as f(x), g(x), h(x), and p(t) using graphs, tables, and algebraic methods. Concepts like slope, point-slope form, and table lookups are used. Suitable for high school students in Grades 9-11.

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