If g(x) = x$^2$ + 3x  find g(x + 1) - g(x)

Analysis

To solve this problem, we need to evaluate the function $g(x)$ at two different points: $x+1$ and $x$, and then find the difference between the two results.

Concept

The function is given as $g(x) = x^2 + 3x$. To find $g(x+1)$, we substitute $(x+1)$ wherever we see $x$ in the original function. After finding both expressions, we subtract $g(x)$ from $g(x+1)$.

Step-by-Step Solution

Step 1: Find $g(x+1)$ Substitute $(x+1)$ into the function $g(x) = x^2 + 3x$: $g(x+1) = (x+1)^2 + 3(x+1)$

Expand the terms: $(x+1)^2 = x^2 + 2x + 1$ $3(x+1) = 3x + 3$

Combine them: $g(x+1) = (x^2 + 2x + 1) + (3x + 3)$ $g(x+1) = x^2 + 5x + 4$

Step 2: Find $g(x+1) - g(x)$ Now, subtract the original function $g(x) = x^2 + 3x$ from the result in Step 1: $[x^2 + 5x + 4] - [x^2 + 3x]$

Distribute the negative sign: $x^2 + 5x + 4 - x^2 - 3x$

Group like terms: $(x^2 - x^2) + (5x - 3x) + 4$ $0 + 2x + 4$ $2x + 4$

Step 3: Factorize the result Factor out 2 from the expression: $2(x + 2)$

Conclusion

Comparing our result $2(x + 2)$ with the given options, it matches option B.

Exam Tip

When dealing with functions like $g(x+a)$, always expand the brackets carefully. A common mistake is to write $(x+1)^2$ as $x^2 + 1$ instead of $x^2 + 2x + 1$. Always remember the identity $(a+b)^2 = a^2 + 2ab + b^2$.