Function g(x) Analysis

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Questions and Answers

What is the value of g(x) when x is not equal to 5?

0
x + 5 (correct)
5
10

At which point does the function g(x) have a point of discontinuity?

x = 0
x = -5
x = 5 (correct)
x = 1

What is the value of the limit of g(x) as x approaches 5?

0
Undefined
10 (correct)
5

For x ≠ 5, how can g(x) be simplified?

g(x) = x + 5 (D) Signup and view all the answers

What type of discontinuity does g(x) have at x = 5?

Removable discontinuity (C) Signup and view all the answers

How is the discontinuity in g(x) removed?

By explicitly defining g(5) = 10 (A) Signup and view all the answers

What is the redefined value of g(5) that makes g(x) continuous?

10 (A) Signup and view all the answers

What is the original form of g(x) for x ≠ 5?

g(x) = (x² - 25) / (x - 5) (A) Signup and view all the answers

What does it mean for g(x) to have a 'hole' at x = 5?

The function is undefined at x=5 (B) Signup and view all the answers

After removing the discontinuity, what is true about g(x)?

g(x) is continuous everywhere (D) Signup and view all the answers

When x = 1, what is g(x) equal to?

0 (C) Signup and view all the answers

How can you determine that $g(x)$ has a removable discontinuity at $x=5$?

Because the limit of $g(x)$ as $x \to 5$ exists, and redefining $g(5)$ consistently removes the discontinuity. (D) Signup and view all the answers

In the original definition of $g(x)$, what value is assigned to $g(5)$?

10 (C) Signup and view all the answers

What algebraic technique is used to simplify the expression $\frac{x^2 - 25}{x - 5}$?

Factoring the numerator (B) Signup and view all the answers

For what values of $x$, other than $x=5$, is the simplified form $g(x)=x+5$ valid?

For all real numbers except $x=5$ (C) Signup and view all the answers

What is the value of $g(x)$ when $x=0$?

5 (A) Signup and view all the answers

What is the significance of the limit existing as $x$ approaches $5$ for the given function?

It suggests that the discontinuity at $x=5$ is removable. (D) Signup and view all the answers

If $g(x)$ were not redefined at $x=5$, what would be the nature of the function at that point?

Undefined, creating a 'hole' (B) Signup and view all the answers

What characteristic of the original function $\frac{x^2 - 25}{x - 5}$ leads to the discontinuity at $x=5$?

The denominator becomes zero. (A) Signup and view all the answers

What is the primary reason for making $g(x)$ continuous everywhere?

To satisfy a specific mathematical condition or requirement. (A) Signup and view all the answers

Flashcards

Point of Discontinuity

A function that is not continuous at a specific point.

Removable Discontinuity

Since the limit exists and matches the redefined value g(5) = 10, the discontinuity is removable.

Limit Value at x=5

The limit of g(x) as x approaches 5 is 10. lim (x->5) g(x) = 10

Simplified g(x)

g(x) = x + 5 for x ≠ 5.

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Continuity Definition

The discontinuity is removed by explicitly defining g(5) = 10, making g(x) continuous everywhere.

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Study Notes

Smart Stella Chidinma, CSC/23/6891 is the author of the notes.

Function definition

g(x) = (x²-25)/(x-5) when x ≠ 5
g(x) = 10 when x = 5

Analysis of g(x)

Considering the function g(x)
When x = 1, initially g(x) = (x²-25)/(x-5) is evaluated as (5²-25)/(5-5) = 0/0.
The function has a point of discontinuity at x = 5.

Simplification for x ≠ 5

For x ≠ 5, g(x) = (x²-25)/(x-5) = ((x-5)(x+5))/(x-5) = x + 5.

Limit Evaluation

Finding the limit as x approaches 5.
For x ≠ 5, g(x) = x + 5.
The limit of g(x) as x approaches 5 is lim (x→5) g(x) = 5 + 5 = 10.

Discontinuity

g(x) has a hole at x = 5 due to the (x²-25)/(x-5) form.
The discontinuity is removable because the limit exists and matches the redefined value g(5) = 10.
The discontinuity is removed by explicitly defining g(5) = 10, making g(x) continuous everywhere.