Determine if the given relations f, g, and h are functions. Provide reasons for your answers.

The relation $f$ is defined as:
$f(a) = s$
$f(b) = u$
$f(c) = t$
The element $d \in A$ is not assigned to any element in $B$. This violates the first rule of functions.

• Is $f$ a function? ❌ No
• Reason: The element $d$ from the domain $A$ is not defined (not assigned an image) in the codomain $B$.

The relation $g$ is defined as:
$g(a) = s$
$g(b) = u$
$g(c) = u$
$g(d) = t$
Every element in $A$ ($a, b, c, d$) is assigned to exactly one element in $B$. For example, $g(b)=u$ and $g(c)=u$ is allowed; it just means two different inputs map to the same output. This satisfies both rules of functions.

• Is $g$ a function? ✅ Yes
• Reason: Every element in the domain $A$ is assigned to exactly one element in the codomain $B$.

The relation $h$ is defined as:
$h(a) = s$
$h(b) = u$
$h(b) = t$
$h(c) = u$
$h(d) = t$
The element $b \in A$ is assigned to two different elements in $B$: $u$ and $t$. This violates the second rule of functions.

• Is $h$ a function? ❌ No
• Reason: The element $b$ from the domain $A$ is assigned to two different images ($u$ and $t$) in the codomain $B$.

Here are the answers:

• $f$ is not a function because $f(d)$ is not defined.
• $g$ is a function because every element of the source set $A$ is assigned an image in the target set $B$, and each element in $A$ has exactly one image.
• $h$ is not a function because an element from the source set, $b$, has two different images ($u$ and $t$) in the target set.