Find the Value of a so that the Function is Continuous Everywhere

TL;DR

Determine the value of the constant 'a' to make the function continuous everywhere.

Transcript

in this problem we have to find the value of the constant a so that the function is continuous everywhere that's the goal so find the value of a so that we can make this function continuous so this is a piecewise function and by themselves the pieces are continuous this is a parabola it's continuous and this is a line and it's continuous but if you... Read More

Key Insights

• 🫥 The function is piecewise, consisting of a parabola and a line.
• 👈 The focus is on making the function continuous at the point 'x = 1'.
• ☺️ One-sided limits are used to evaluate the behavior of the function before and after 'x = 1'.
• ⛔ By equating the one-sided limits, the value of 'a' that achieves continuity is found.
• ❓ The constant 'a' is determined to be 13 to make the function continuous everywhere.
• 😑 Inequalities in the function guide the selection of relevant expressions for one-sided limits.
• 🥳 The concept of continuity plays a crucial role in connecting different parts of the piecewise function.

Questions & Answers

Q: What is the objective of the problem?

The objective is to find the value of constant 'a' to make the given function continuous everywhere.

Q: Why is it necessary to consider one-sided limits?

One-sided limits help determine the behavior of the function as it approaches a specific point, in this case, 'x = 1'.

Q: How do we calculate the limit from the left side of 'x = 1'?

We use the bottom piece of the function, substituting 'f(x)' with 'ax - 5' and evaluate the limit by plugging in 'x = 1'.

Q: What equation represents the limit from the right side of 'x = 1'?

The top piece of the function, with 'f(x)' replaced by '8x^2', represents the limit from the right side of 'x = 1'.

Q: Why do we set the limits equal to each other?

By equating the limits from both sides, we ensure that the function is continuous at 'x = 1' and find the value of 'a' to achieve that continuity.

Summary & Key Takeaways

• The goal is to find the value of 'a' so that the function is continuous.

• The function is piecewise, with a parabola and a line.

• The focus is on ensuring continuity at the point 'x = 1' by considering one-sided limits.