Let f:R→R be defined by f(x)=−8x−3. Is f a linear transformation?

f(x+y)= 
f(x)+f(y)= 
Does f(x+y)=f(x)+f(y) for all x,y∈R? 


f(cx)= 
c(f(x))= 
Does f(cx)=c(f(x)) for all c,x∈R?  



Is f a linear transformation?  

To determine if \( f: \mathbb{R} \rightarrow \mathbb{R} \), defined by \( f(x) = -8x - 3 \), is a linear transformation, we need to check two properties for all \( x, y \in \mathbb{R} \) and for any scalar \( c \in \mathbb{R} \):

1. **Additivity**: \( f(x + y) = f(x) + f(y) \)
2. **Homogeneity (scalar multiplication)**: \( f(cx) = c \cdot f(x) \)

Let's go through each property.

### 1. Additivity
We calculate \( f(x + y) \) and \( f(x) + f(y) \) separately.

- **\( f(x + y) \):**
  \[
  f(x + y) = -8(x + y) - 3 = -8x - 8y - 3
  \]

- **\( f(x) + f(y) \):**
  \[
  f(x) = -8x - 3, \quad f(y) = -8y - 3
  \]
  \[
  f(x) + f(y) = (-8x - 3) + (-8y - 3) = -8x - 8y - 6
  \]

Since \( f(x + y) = -8x - 8y - 3 \) and \( f(x) + f(y) = -8x - 8y - 6 \), we see that:
\[
f(x + y) \neq f(x) + f(y)
\]
Therefore, **the additivity property fails** for \( f(x) \).

### 2. Homogeneity (Scalar Multiplication)
We calculate \( f(cx) \) and \( c \cdot f(x) \) separately.

- **\( f(cx) \):**
  \[
  f(cx) = -8(cx) - 3 = -8cx - 3
  \]

- **\( c \cdot f(x) \):**
  \[
  f(x) = -8x - 3
  \]
  \[
  c \cdot f(x) = c(-8x - 3) = -8cx - 3c
  \]

Since \( f(cx) = -8cx - 3 \) and \( c \cdot f(x) = -8cx - 3c \), we see that:
\[
f(cx) \neq c \cdot f(x)
\]
Thus, **the homogeneity property also fails** for \( f(x) \).

### Conclusion
Since \( f(x) = -8x - 3 \) does not satisfy either the additivity or homogeneity properties, **\( f \) is not a linear transformation**.

---

Would you like further explanation on any step? 

Here are 5 questions to deepen your understanding:

1. What are some examples of functions that are linear transformations?
2. How would the function \( f(x) = -8x \) differ in terms of linearity?
3. Can a transformation that does not satisfy additivity still satisfy homogeneity?
4. What impact does the constant term \(-3\) have on the linearity of \( f(x) \)?
5. How would we define the image of \( f(x) \) in this context?

**Tip:** For a function to be a linear transformation, it must map the origin (0) to itself, i.e., \( f(0) = 0 \).

Let f: R→R be defined by f(x) = −8x − 3. Is f a linear transformation?

f(x + y) =
f(x) + f(y) =
Does f(x + y) = f(x) + f(y) for all x, y ∈ R?

f(cx) =
c(f(x)) =
Does f(cx) = c(f(x)) for all c, x ∈ R?

Is f a linear transformation?

Explore whether the function f(x) = -8x - 3 qualifies as a linear transformation. Analyze key properties of linear transformations, including additivity and homogeneity, and learn why f fails to meet the criteria. Suitable for advanced high school and undergraduate math students.

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