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a) Let [tex]$f: \mathbb{R} \rightarrow \mathbb{R}$[/tex] be defined by [tex]$f(x) = ax + b$[/tex], [tex][tex]$x \in \mathbb{R}$[/tex][/tex]. If [tex]$f(1) = 5$[/tex] and [tex]$f(-4) = -5$[/tex], find the values of [tex][tex]$a$[/tex][/tex] and [tex]$b$[/tex]. Also, find the functions [tex]f(n)[/tex] and [tex]f(-1)[/tex].



Answer :

GinnyAnswer
Let's solve for the coefficients [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that the function [tex]\(f(x) = ax + b\)[/tex] satisfies the given conditions:
1. [tex]\(f(1) = 5\)[/tex]
2. [tex]\(f(-4) = -5\)[/tex]

We start by plugging the values into the given function:

Step 1: Formulate the first equation using [tex]\(f(1) = 5\)[/tex]:
[tex]\[ f(1) = a \cdot 1 + b = 5 \][/tex]
This simplifies to:
[tex]\[ a + b = 5 \][/tex]

Step 2: Formulate the second equation using [tex]\(f(-4) = -5\)[/tex]:
[tex]\[ f(-4) = a \cdot (-4) + b = -5 \][/tex]
This simplifies to:
[tex]\[ -4a + b = -5 \][/tex]

Now we have a system of linear equations:
1. [tex]\( a + b = 5 \)[/tex]
2. [tex]\(-4a + b = -5\)[/tex]

Step 3: Solve the system of equations

We can use the method of elimination to solve for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]. First, we subtract the second equation from the first to eliminate [tex]\(b\)[/tex]:

[tex]\[ (a + b) - (-4a + b) = 5 - (-5) \][/tex]
[tex]\[ a + b + 4a - b = 5 + 5 \][/tex]
[tex]\[ 5a = 10 \][/tex]
[tex]\[ a = 2 \][/tex]

Step 4: Determine [tex]\(b\)[/tex] using the value of [tex]\(a\)[/tex]:

Substitute [tex]\(a = 2\)[/tex] back into the first equation:
[tex]\[ a + b = 5 \][/tex]
[tex]\[ 2 + b = 5 \][/tex]
[tex]\[ b = 3 \][/tex]

So we have found the values:
[tex]\[ a = 2 \][/tex]
[tex]\[ b = 3 \][/tex]

Step 5: Write the function [tex]\(f(x)\)[/tex]:

Given [tex]\(a\)[/tex] and [tex]\(b\)[/tex], the function [tex]\(f(x)\)[/tex] becomes:
[tex]\[ f(x) = 2x + 3 \][/tex]

Step 6: Determine [tex]\(f(-1)\)[/tex]:

Substitute [tex]\(x = -1\)[/tex] into the function:
[tex]\[ f(-1) = 2(-1) + 3 \][/tex]
[tex]\[ f(-1) = -2 + 3 \][/tex]
[tex]\[ f(-1) = 1 \][/tex]

Summary:

- The values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are [tex]\(a = 2\)[/tex] and [tex]\(b = 3\)[/tex].
- The function [tex]\(f(x)\)[/tex] is [tex]\(f(x) = 2x + 3\)[/tex].
- The value of [tex]\(f(-1)\)[/tex] is [tex]\(f(-1) = 1\)[/tex].

So, we have:
[tex]\[ a = 2 \][/tex]
[tex]\[ b = 3 \][/tex]
[tex]\[ f(x) = 2x + 3 \][/tex]
[tex]\[ f(-1) = 1 \][/tex]

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