Question

Yes or No? If No, give a reason. Let f be a function.

(a) Is it possible that $f\left(1\right)=5$and $f\left(2\right)=5$?

(b) Is it possible that $f\left(1\right)=5$ and $f\left(1\right)=6$?

Short answer

(a) Yes

(b) NO

Step-by-step solution

Part a. Step 1. Use the definition of functions.

Function, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

A function is also defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x,y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$.

This means that if the object x is in the set of inputs (called the domain) then a function f will map the object x to exactly one object f(x) in the set of possible outputs (called the range).

Part a. Step 2. Investigate the statements.

We are given the following function f such that$f\left(1\right)=5$ and $f\left(2\right)=5$ , since for two different x values functions values can be same this is possible.

Part a. Step 3. Evaluate the results.

The this statement completes the criterion for a function thus, it answer should be “Yes”.

Part b. Step 1. Use the definition of functions.

Function, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

A function is also defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x,y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$.

This means that if the object x is in the set of inputs (called the domain) then a function f will map the object x to exactly one object f(x) in the set of possible outputs (called the range).

Part b. Step 2. Investigate the statements.

We are given the following function f such that $f\left(1\right)=5$ and $f\left(1\right)=6$ , since for same x values functions y- values cannot be same this is not possible.

Part b. Step 3. Evaluate the results.

The this statement does not completes the criterion for a function thus, it answer should be “No”.