Sure, let's work through the problems step-by-step.
### Part (a) Evaluate [tex]\( k(2) \)[/tex]
We are given the function [tex]\( k(t) = 2t - 1 \)[/tex]. To evaluate [tex]\( k(2) \)[/tex], we substitute [tex]\( t = 2 \)[/tex] into the function:
[tex]\[ k(2) = 2(2) - 1 \][/tex]
First, do the multiplication:
[tex]\[ 2 \times 2 = 4 \][/tex]
Then, subtract 1:
[tex]\[ 4 - 1 = 3 \][/tex]
So, [tex]\( k(2) = 3 \)[/tex].
### Part (b) Solve [tex]\( k(t) = 7 \)[/tex]
We need to find the value of [tex]\( t \)[/tex] such that [tex]\( k(t) = 7 \)[/tex]. Using the function definition [tex]\( k(t) = 2t - 1 \)[/tex], we set up the equation:
[tex]\[ 2t - 1 = 7 \][/tex]
To solve for [tex]\( t \)[/tex], first add 1 to both sides of the equation:
[tex]\[ 2t - 1 + 1 = 7 + 1 \][/tex]
This simplifies to:
[tex]\[ 2t = 8 \][/tex]
Next, divide both sides by 2:
[tex]\[ t = \frac{8}{2} \][/tex]
This simplifies to:
[tex]\[ t = 4 \][/tex]
Therefore, when [tex]\( k(t) = 7 \)[/tex], [tex]\( t = 4 \)[/tex].
### Summary
- For part (a), [tex]\( k(2) = 3 \)[/tex].
- For part (b), the solution of [tex]\( k(t) = 7 \)[/tex] is [tex]\( t = 4 \)[/tex].
