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Suppose you have a linear transformation [tex]f: \mathbb{R} \rightarrow \mathbb{R}[/tex], where [tex]f(3) = 9[/tex] and [tex]f(5) = 15[/tex].

Use the addition property to compute [tex]f(8)[/tex] and [tex]f(13)[/tex]. Then find [tex]f(-3)[/tex].

Choices:
A. [tex]-9[/tex]
B. [tex]-24[/tex]
C. [tex]-15[/tex]
D. [tex]0[/tex]

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].

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