Constant Function

 

Let $f: A \to B$ be a given function. If every element in the domain $A$ maps to the exact same image in the codomain $B$, then this function is called a constant function.

That is, for all $x \in A$, $f(x) = c$ where $c \in B$.

The total number of unique constant functions that can be defined from $A$ to $B$ is equal to the cardinality of the codomain, $s(B)$.

 

Example:

 

The function $f: A \to B$ is defined by the rule $f(x) = c$.

\[
f(1) = c, \quad f(2) = c, \quad f(3) = c, \quad f(4) = c
\]

In this case, exactly $s(B) = 4$ different constant functions can be defined.

 

QUESTION 20

 

Given that the function $f: \mathbb{R} \to \mathbb{R}$, defined by
\[
f(x) = \frac{(a – 1)x^2 – 3}{ax^2 – 1}
\]
is a constant function, what is the value of $a$?

\[
\text{A)} -1 \quad
\text{B) } -\frac{1}{2} \quad
\text{C) } 0 \quad
\text{D) } \frac{1}{2} \quad
\text{E) } 1
\]

 

Solution:

 

For $f$ to be a constant function, it must satisfy $f(x) = c$ for some constant $c \in \mathbb{R}$.

\[
f(x) = \frac{(a – 1)x^2 – 3}{ax^2 – 1} = c
\]

\[
\Rightarrow (a – 1)x^2 – 3 = acx^2 – c
\]

By equating the coefficients of like terms, we get:

\[
\Rightarrow -3 = -c \quad \text{and} \quad a – 1 = ac
\]

\[ \Rightarrow c = 3 \quad \text{and} \quad a – 1 = 3a \]

\[ \Rightarrow 2a = -1 \Rightarrow a = -\frac{1}{2} \]

 

\(\textbf{Correct Answer: B} \)

 

Graph of a Constant Function:

 

The graph of a constant function $f: \mathbb{R} \to \mathbb{R}, \;\; f(x) = c$ is represented by a horizontal line:

 

Example:

 

The constant function
\[
f: \mathbb{R} \to \mathbb{R}, \quad f(x) = -2
\]
is illustrated above. For this function, $f(10) = -2$ and $f(100) = -2$.