Question

Let \(f\) and \(g\) be the functions from the set of integers to the set of integers defined by \[ f(x)=2x+3 \] and \[ g(x)=3x+2. \] What is the composition of \(f\) and \(g\)?

• A. \(6x+7\)
• B. \(6x+11\)
• C. \(8x+11\)
• D. \(8x+7\)

Hint:

To find a composition of functions: \[ (f\circ g)(x)=f(g(x)). \] Always substitute the entire expression \(g(x)\) into \(f(x)\), and then simplify carefully.

Answer 1

The Correct Option is A

Concept: The composition of two functions means applying one function after the other. If \(f:A\rightarrow B\) and \(g:B\rightarrow C\), then the composition of \(f\) and \(g\) is denoted by \[ (g\circ f)(x)=g(f(x)). \] Similarly, \[ (f\circ g)(x)=f(g(x)). \] To find the composition, we first evaluate the inner function and then substitute its value into the outer function. Function composition is widely used in algebra, calculus, and discrete mathematics to combine two transformations into a single transformation.

Step 1: Write the given functions.
We are given \[ f(x)=2x+3 \] and \[ g(x)=3x+2. \] The composition of \(f\) and \(g\) means \[ (f\circ g)(x)=f(g(x)). \]

Step 2: Find \(g(x)\).
Substituting directly, \[ g(x)=3x+2. \] This expression will now be used as the input of function \(f\).

Step 3: Substitute \(g(x)\) into \(f(x)\).
Since \[ f(x)=2x+3, \] replacing \(x\) by \(g(x)\), \[ f(g(x)) = 2(3x+2)+3. \]

Step 4: Simplify the expression.
Expanding the bracket, \[ f(g(x)) = 6x+4+3. \] Combining the constant terms, \[ f(g(x)) = 6x+7. \]

Step 5: Write the final answer.
Therefore, \[ (f\circ g)(x)=6x+7. \] Hence the correct option is \[ \boxed{(A)\;6x+7}. \]