Answer: k = 4
Reason
Define functions g(x) and h(x) such that:
The two functions make up the piecewise function f(x) as shown in the diagram you posted.
- If x < 1, then f(x) = g(x).
- If
, then f(x) = h(x).
In order for the curves of g(x) and h(x) to connect, we need g(1) = h(1) to be true. The value x = 1 is where the junction point is.
Plug x = 1 into each.
g(x) = 5x+k
g(1) = 5(1)+k
g(1) = 5+k
and
h(x) = 9/x
h(1) = 9/1
h(1) = 9
So,
g(1) = h(1)
5+k = 9
k = 9-5
k = 4 which is the final answer.
This is the value of k where g(1) = h(1) is true and it's the value of k that makes pieces g(x) and h(x) to join up. The diagram below confirms this. I used GeoGebra to make the graph. But Desmos is a great alternative choice.