Let $g\left(x\right)=\begin{cases}x^2+x & \text{if }x<1\\ a & \text{if }x=1\\ 3x+5 & \text{if }x>1\end{cases}$
b. Determine the value of $a$ for which $g$ is continuous from the right at $1$. 

Verified step by step guidance

1. To determine the value of 'a' for which the function g(x) is continuous from the right at x = 1, we need to ensure that the right-hand limit of g(x) as x approaches 1 is equal to g(1).
2. The right-hand limit of g(x) as x approaches 1 is found by considering the expression for g(x) when x > 1, which is 3x + 5.
3. Calculate the right-hand limit: \( \lim_{{x \to 1^+}} g(x) = \lim_{{x \to 1^+}} (3x + 5) \).
4. Evaluate this limit by substituting x = 1 into the expression 3x + 5, which gives 3(1) + 5.
5. For g(x) to be continuous from the right at x = 1, set the right-hand limit equal to g(1), which is 'a'. Therefore, solve the equation 3(1) + 5 = a to find the value of 'a'.

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function g(x) has three distinct cases depending on whether x is less than, equal to, or greater than 1. Understanding how to evaluate piecewise functions is crucial for determining their properties, such as continuity.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point from both sides equals the function's value at that point. For g(x) to be continuous at x=1, the limit as x approaches 1 from the left must equal the limit as x approaches 1 from the right, and both must equal g(1). This concept is essential for solving the problem.

Limits

Limits describe the behavior of a function as the input approaches a certain value. In this context, we need to find the left-hand limit (as x approaches 1 from values less than 1) and the right-hand limit (as x approaches 1 from values greater than 1) of g(x). Evaluating these limits will help determine the appropriate value of a that ensures continuity at x=1.