Inputing Values into a Function

In mathematics, we often assign numerical values to variables in equations or mathematical expressions. A function can be thought of as a machine: you input a value (called the input), the function processes it according to a specific rule, and produces a result (called the output). The input is usually represented by \(x)\, and the output by f(x)f(x) or yy.

Understanding Variables and Substitution

A variable is a placeholder for an unknown number, often called the "unknown". When we replace a variable with a specific value, we refer to this process as substitution.

  • If an equation contains only one variable, we can solve for its value by isolating it.
  • If an equation contains multiple variables, there can be multiple solutions. Each value substituted affects the remaining variables in the equation.

When we talk about inputting values into a function, we are essentially substituting the variables in a mathematical expression or equation with specific numerical values to evaluate the result.
Often "solving an equation" is actually finding the values of the variables inside it.

For example

Suppose we have three variables, two of which have known values:

X=3 X=3

Y=2 Y=2

Z=? Z=\text{?}

We are also given the equation:

X2+Y=Z X^2+Y=Z

Remember! when facing this kind of questions, you want to try and find the values of the variables.
To solve the problem, we will first substitute the known values into the equation:

32+2=Z 3^2+2=Z

Simplify and solve:

9+2=Z9+2=Z

Z=11Z=11

By substituting the known values and performing the necessary operations, we were able to isolate and calculate the value of ZZ

Answer: Z=11 Z=11

X Y Z By assigning the numerical value, the general form becomes a particular case

Suggested Topics to Practice in Advance

  1. Ways to Represent a Function
  2. Representing a Function Verbally and with Tables
  3. Graphical Representation of a Function
  4. Algebraic Representation of a Function
  5. Notation of a Function
  6. Rate of Change of a Function
  7. Variation of a Function
  8. Rate of change represented with steps in the graph of the function
  9. Rate of change of a function represented graphically
  10. Constant Rate of Change
  11. Variable Rate of Change
  12. Rate of Change of a Function Represented by a Table of Values
  13. Functions for Seventh Grade
  14. Increasing and Decreasing Intervals (Functions)
  15. Increasing functions
  16. Decreasing function
  17. Constant Function
  18. Decreasing Interval of a function
  19. Increasing Intervals of a function

Practice Inputing Values into a Function

Examples with solutions for Inputing Values into a Function

Exercise #1

6x+5=1 \frac{6}{x+5}=1

What is the field of application of the equation?

Video Solution

Step-by-Step Solution

To solve this problem, we will determine the domain, or field of application, of the equation 6x+5=1 \frac{6}{x+5} = 1 .

Step-by-step solution:

  • Step 1: Identify the denominator. In the given equation, the denominator is x+5 x+5 .
  • Step 2: Determine when the denominator is zero. Solve for x x by setting x+5=0 x+5 = 0 .
  • Step 3: Solve the equation: x+5=0 x+5 = 0 gives x=5 x = -5 .
  • Step 4: Exclude this value from the domain. The domain is all real numbers except x=5 x = -5 .

Therefore, the field of application of the equation is all real numbers except where x=5 x = -5 .

Thus, the domain is x5 x \neq -5 .

Answer

x5 x\operatorname{\ne}-5

Exercise #2

x+y:32x+6=4 \frac{x+y:3}{2x+6}=4

What is the field of application of the equation?

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps to find the domain:

  • Step 1: Recognize that the expression x+y:32x+6=4\frac{x+y:3}{2x+6}=4 involves a fraction. The denominator 2x+62x + 6 must not be zero, as division by zero is undefined.
  • Step 2: Set the denominator equal to zero and solve for xx to find the values that must be excluded: 2x+6=02x + 6 = 0.
  • Step 3: Solve 2x+6=02x + 6 = 0:
    • 2x+6=02x + 6 = 0
    • 2x=62x = -6
    • x=3x = -3
  • Step 4: Conclude that the domain of the function excludes x=3x = -3, meaning x3x \neq -3.

Thus, the domain of the given expression is all real numbers except x=3x = -3. This translates to:

x3 x\operatorname{\ne}-3

Answer

x3 x\operatorname{\ne}-3

Exercise #3

3x:4y+6=6 \frac{3x:4}{y+6}=6

What is the field of application of the equation?

Video Solution

Step-by-Step Solution

To determine the field of application of the equation 3x:4y+6=6\frac{3x:4}{y+6}=6, we must identify values of yy for which the equation is defined.

  • The denominator of the given expression is y+6y + 6. In order for the expression to be defined, the denominator cannot be zero.
  • This leads us to solve the equation y+6=0y + 6 = 0.
  • Solving y+6=0y + 6 = 0 gives us y=6y = -6.
  • This means y=6y = -6 would make the denominator zero, thus the expression would be undefined for this value.

Therefore, the field of application, or the domain of the equation, is all real numbers except y=6y = -6.

We must conclude that y6 y \neq -6 .

Comparing with the provided choices, the correct answer is choice 3: y6 y \neq -6 .

Answer

y6 y\operatorname{\ne}-6

Exercise #4

22(2x1)=30 22(\frac{2}{x}-1)=30

What is the domain of the equation above?

Video Solution

Step-by-Step Solution

To find the domain of the given function 22(2x1)=30 22\left(\frac{2}{x} - 1\right) = 30 , follow these steps:

  • Identify critical terms: The term 2x\frac{2}{x} is undefined when x=0 x = 0 because division by zero is undefined.
  • We need to exclude x=0 x = 0 from the domain to ensure the function remains defined.
  • The correct domain for the equation is all real numbers except x=0 x = 0 .

Thus, the domain of the equation is x0 x \neq 0 .

Therefore, the solution to the problem is x0 x \neq 0 .

Answer

x≠0

Exercise #5

2x+6x=18 2x+\frac{6}{x}=18

What is the domain of the above equation?

Video Solution

Step-by-Step Solution

To solve this problem and find the domain for the expression 2x+6x2x + \frac{6}{x}, we apply the following steps:

  • Step 1: Identify when the fraction 6x\frac{6}{x} is undefined. This occurs when the denominator xx equals zero.
  • Step 2: To find the restriction, set the denominator equal to zero: x=0x = 0.
  • Step 3: Solve for xx to find the values excluded from the domain. Here, x0x \neq 0.

Since 6x\frac{6}{x} is undefined for x=0x = 0, the value x=0x = 0 must be excluded from the domain.
Hence, the domain of the equation is all real numbers except zero.

Therefore, the solution to the problem, indicating the domain of the expression, is x0 x \neq 0 .

Answer

x≠0

Exercise #6

2x3=4x 2x-3=\frac{4}{x}

What is the domain of the exercise?

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the fraction's denominator.

  • Step 2: Determine where this denominator equals zero.

  • Step 3: Exclude this value from the domain.

Now, let's work through each step:

Step 1: The given equation is 2x3=4x 2x - 3 = \frac{4}{x} . Notice that the fraction 4x\frac{4}{x} has a denominator of xx.

Step 2: Set the denominator equal to zero to determine where it is undefined.

xamp;=0 \begin{aligned} x &= 0 \end{aligned}

Step 3: Since the expression is undefined at x=0x = 0, we must exclude this value from the domain.

Therefore, the domain of the expression is all real numbers except 0, formally stated as x0 x \neq 0 .

The correct solution to the problem is: x ≠ 0.

Answer

x≠0

Exercise #7

What is the domain of the exercise?

5x+82x6=30 \frac{5x+8}{2x-6}=30

Video Solution

Step-by-Step Solution

To find the domain of the expression 5x+82x6=30\frac{5x+8}{2x-6} = 30, we need to identify values of xx that make the denominator of the fraction zero.

Step 1: Identify the denominator of the fraction, which is 2x62x - 6.

Step 2: Set the denominator equal to zero to find the values to exclude:

  • Solve the equation 2x6=02x - 6 = 0.
  • Add 6 to both sides: 2x=62x = 6.
  • Divide both sides by 2: x=3x = 3.

Therefore, x=3x = 3 is the value that makes the denominator zero, so it must be excluded from the domain.

Given the choices, the correct answer is x3x \neq 3.

Therefore, the domain of the expression is all real numbers except x=3x = 3.

This implies that the correct choice is:

x3 x \neq 3

Answer

x≠3

Exercise #8

Look at the following function:

2010x5 \frac{20}{10x-5}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To determine the domain of the function 2010x5 \frac{20}{10x-5} , we need to ensure that the denominator is not zero.

  • Step 1: Identify the denominator, which is 10x5 10x - 5 .
  • Step 2: Set the denominator equal to zero and solve for x x . This gives us the equation:

10x5=0 10x - 5 = 0

  • Step 3: Add 5 to both sides of the equation:

10x=5 10x = 5

  • Step 4: Divide both sides by 10 to isolate x x :

x=510 x = \frac{5}{10}

  • Step 5: Simplify the fraction:

x=12 x = \frac{1}{2}

This means that the function is undefined at x=12 x = \frac{1}{2} . Therefore, the domain of the function is all real numbers except x=12 x = \frac{1}{2} .

Therefore, the domain of the function is x12 x \ne \frac{1}{2} .

Answer

x12 x\ne\frac{1}{2}

Exercise #9

Look at the following function:

2x+23x1 \frac{2x+2}{3x-1}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To find the domain of the function 2x+23x1 \frac{2x + 2}{3x - 1} , we must ensure that the function is defined for all values of x x except where the denominator is zero.

Follow these steps to determine the domain:

  • Step 1: Identify the denominator of the function. The denominator is 3x1 3x - 1 .
  • Step 2: Set the denominator equal to zero and solve for x x : 3x1=0 3x - 1 = 0
  • Step 3: Solve the equation 3x1=0 3x - 1 = 0 : 3x=1 3x = 1 x=13 x = \frac{1}{3}
  • Step 4: State the domain. Since the function is undefined at x=13 x = \frac{1}{3} , the domain consists of all real numbers except x=13 x = \frac{1}{3} .

The domain of the function 2x+23x1 \frac{2x + 2}{3x - 1} is therefore all real numbers x x such that x13 x \neq \frac{1}{3} .

Thus, the solution is: x13 x \ne \frac{1}{3} .

Answer

x13 x\ne\frac{1}{3}

Exercise #10

Consider the following function:

3x+42x1 \frac{3x+4}{2x-1}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To determine the domain of the function 3x+42x1 \frac{3x+4}{2x-1} , follow these steps:

  • Step 1: Identify the denominator of the rational function, which is 2x1 2x-1 .
  • Step 2: Set the denominator equal to zero to find the values of x x that make the function undefined:
    2x1=0 2x - 1 = 0 .
  • Step 3: Solve for x x :
    Add 1 to both sides: 2x=1 2x = 1 .
    Divide both sides by 2: x=12 x = \frac{1}{2} .

The value x=12 x = \frac{1}{2} makes the denominator zero, which means the function 3x+42x1 \frac{3x+4}{2x-1} is undefined at x=12 x = \frac{1}{2} . Therefore, this value must be excluded from the domain.

The domain of the function is all real numbers except x=12 x = \frac{1}{2} .

Therefore, the solution to the problem is x12 x \ne \frac{1}{2} .

Answer

x12 x\ne\frac{1}{2}

Exercise #11

Look at the following function:

10x35x3 \frac{10x-3}{5x-3}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To find the domain of the function 10x35x3\frac{10x-3}{5x-3}, we'll follow these steps:

  • Identify the denominator: B(x)=5x3B(x) = 5x - 3.
  • Set the denominator equal to zero: 5x3=05x - 3 = 0.
  • Solve for xx: Add 3 to both sides, getting 5x=35x = 3. Then, divide by 5: x=35x = \frac{3}{5}.
  • Conclude that the domain is all real numbers except x=35x = \frac{3}{5}, since this makes the denominator zero.

Therefore, the domain of the function is all real numbers except x35 x\ne\frac{3}{5} .

Answer

x35 x\ne\frac{3}{5}

Exercise #12

Look at the following function:

2x+29x+6 \frac{2x+2}{9x+6}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To solve this problem, we will determine the domain of the rational function by following these steps:

  • Step 1: Identify the denominator of the function, which is 9x+6 9x + 6 .
  • Step 2: Set the denominator equal to zero to find values of x x that need to be excluded from the domain: 9x+6=0 9x + 6 = 0 .
  • Step 3: Solve the equation 9x+6=0 9x + 6 = 0 for x x .
  • Step 4: To solve, subtract 6 from both sides to get 9x=6 9x = -6 .
  • Step 5: Divide each side by 9 to solve for x x , resulting in x=23 x = -\frac{2}{3} .
  • Step 6: The domain of the function excludes the value x=23 x = -\frac{2}{3} since it makes the denominator zero.

Thus, the domain of the given function is all real numbers except x=23 x = -\frac{2}{3} , expressed as x23 x \ne -\frac{2}{3} .

Therefore, the correct choice for the domain is: x23 x\ne-\frac{2}{3} .

Answer

x23 x\ne-\frac{2}{3}

Exercise #13

Given the following function:

128x4 \frac{12}{8x-4}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To find the domain of the function 128x4 \frac{12}{8x-4} , we must determine when the denominator equals zero and exclude these values.

Step 1: Set the denominator equal to zero and solve for x x :

8x4=0 8x - 4 = 0

Step 2: Solve the equation 8x4=0 8x - 4 = 0 for x x :

Add 4 to both sides: 8x=4 8x = 4

Divide both sides by 8: x=48=12 x = \frac{4}{8} = \frac{1}{2}

Step 3: The value x=12 x = \frac{1}{2} is where the denominator becomes zero, so this value is excluded from the domain.

Therefore, the domain of the function is all real numbers except x=12 x = \frac{1}{2} .

The domain of the function is x12\boxed{x \ne \frac{1}{2}}.

Answer

x12 x\ne\frac{1}{2}

Exercise #14

Look the following function:

15x4 \frac{1}{5x-4}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To determine the domain of the function 15x4 \frac{1}{5x-4} , we need to find the values of x x for which the function is undefined. This occurs when the denominator equals zero:

First, set the denominator equal to zero:
5x4=0 5x - 4 = 0

Next, solve for x x :
5x=4 5x = 4
x=45 x = \frac{4}{5}

The function is undefined at x=45 x = \frac{4}{5} . Therefore, the domain of the function includes all real numbers except x=45 x = \frac{4}{5} .

In mathematical notation, the domain is:
x45 x \ne \frac{4}{5} .

This matches choice 3 among the given options.

Answer

x45 x\ne\frac{4}{5}

Exercise #15

Given the following function:

2421x7 \frac{24}{21x-7}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To determine the domain of the function 2421x7 \frac{24}{21x-7} , we need to ensure that the denominator is not equal to zero.

Step 1: Set the denominator equal to zero and solve for x x :

  • 21x7=0 21x - 7 = 0

  • 21x=7 21x = 7

  • x=721 x = \frac{7}{21}

  • x=13 x = \frac{1}{3}

The function is undefined when x=13 x = \frac{1}{3} because it would cause division by zero.

Step 2: The domain of the function is all real numbers except x=13 x = \frac{1}{3} .

Therefore, the domain of the function is all x x such that x13 x \neq \frac{1}{3} .

Thus, the correct answer is x13 \boxed{ x\ne\frac{1}{3}} .

Answer

x13 x\ne\frac{1}{3}

Topics learned in later sections

  1. Domain of a Function
  2. Indefinite integral