Answer :
Let's solve the problem step by step:
We have a linear transformation [tex]\( f: \mathbb{R} \rightarrow \mathbb{R} \)[/tex] given by the values [tex]\( f(3) = 9 \)[/tex] and [tex]\( f(5) = 15 \)[/tex].
### Step 1: Determine the Linear Transformation Function
A linear transformation in one variable can be represented as:
[tex]\[ f(x) = mx + c \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
#### Calculate the Slope ([tex]\( m \)[/tex])
The slope [tex]\( m \)[/tex] can be found using the formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Given points are [tex]\((3, 9)\)[/tex] and [tex]\((5, 15)\)[/tex], we calculate:
[tex]\[ m = \frac{15 - 9}{5 - 3} = \frac{6}{2} = 3 \][/tex]
#### Find the Y-Intercept ([tex]\( c \)[/tex])
Now, using one of the points, for instance [tex]\((3, 9)\)[/tex], we can find [tex]\( c \)[/tex]:
[tex]\[ 9 = 3 \cdot 3 + c \][/tex]
[tex]\[ 9 = 9 + c \][/tex]
[tex]\[ c = 0 \][/tex]
The linear transformation function is:
[tex]\[ f(x) = 3x + 0 = 3x \][/tex]
### Step 2: Calculate [tex]\( f(8) \)[/tex] and [tex]\( f(13) \)[/tex]
Using the function [tex]\( f(x) = 3x \)[/tex]:
- For [tex]\( f(8) \)[/tex]:
[tex]\[ f(8) = 3 \cdot 8 = 24 \][/tex]
- For [tex]\( f(13) \)[/tex]:
[tex]\[ f(13) = 3 \cdot 13 = 39 \][/tex]
### Step 3: Calculate [tex]\( f(-3) \)[/tex]
- For [tex]\( f(-3) \)[/tex]:
[tex]\[ f(-3) = 3 \cdot (-3) = -9 \][/tex]
Thus, the values are:
- [tex]\( f(8) = 24 \)[/tex]
- [tex]\( f(13) = 39 \)[/tex]
- [tex]\( f(-3) = -9 \)[/tex]
Finally, from the given options for [tex]\( f(-3) \)[/tex], the correct answer is [tex]\(-9\)[/tex].
We have a linear transformation [tex]\( f: \mathbb{R} \rightarrow \mathbb{R} \)[/tex] given by the values [tex]\( f(3) = 9 \)[/tex] and [tex]\( f(5) = 15 \)[/tex].
### Step 1: Determine the Linear Transformation Function
A linear transformation in one variable can be represented as:
[tex]\[ f(x) = mx + c \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
#### Calculate the Slope ([tex]\( m \)[/tex])
The slope [tex]\( m \)[/tex] can be found using the formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Given points are [tex]\((3, 9)\)[/tex] and [tex]\((5, 15)\)[/tex], we calculate:
[tex]\[ m = \frac{15 - 9}{5 - 3} = \frac{6}{2} = 3 \][/tex]
#### Find the Y-Intercept ([tex]\( c \)[/tex])
Now, using one of the points, for instance [tex]\((3, 9)\)[/tex], we can find [tex]\( c \)[/tex]:
[tex]\[ 9 = 3 \cdot 3 + c \][/tex]
[tex]\[ 9 = 9 + c \][/tex]
[tex]\[ c = 0 \][/tex]
The linear transformation function is:
[tex]\[ f(x) = 3x + 0 = 3x \][/tex]
### Step 2: Calculate [tex]\( f(8) \)[/tex] and [tex]\( f(13) \)[/tex]
Using the function [tex]\( f(x) = 3x \)[/tex]:
- For [tex]\( f(8) \)[/tex]:
[tex]\[ f(8) = 3 \cdot 8 = 24 \][/tex]
- For [tex]\( f(13) \)[/tex]:
[tex]\[ f(13) = 3 \cdot 13 = 39 \][/tex]
### Step 3: Calculate [tex]\( f(-3) \)[/tex]
- For [tex]\( f(-3) \)[/tex]:
[tex]\[ f(-3) = 3 \cdot (-3) = -9 \][/tex]
Thus, the values are:
- [tex]\( f(8) = 24 \)[/tex]
- [tex]\( f(13) = 39 \)[/tex]
- [tex]\( f(-3) = -9 \)[/tex]
Finally, from the given options for [tex]\( f(-3) \)[/tex], the correct answer is [tex]\(-9\)[/tex].
