Tips for Solving Function Rule Problems

Oct 13, 2025 by thesweetdesigns.com 62 views

Understanding the Function Rule

The given function rule is $f(x) = \frac{x}{86}$ . This function takes an input value $x$ and returns an output value that is the result of dividing $x$ by 86.

Applying the Function Rule to Find f(-86)

To find $f(-86)$ , we need to substitute $-86$ into the function rule in place of $x$ . This means we will replace every instance of $x$ with $-86$ and then simplify the resulting expression.

Step 1: Substitute -86 into the Function Rule

$f(-86) = \frac{-86}{86}$

Step 2: Simplify the Expression

When we divide a negative number by a positive number, the result is always negative. In this case, we are dividing $-86$ by $86$ , which is a positive number.

$\frac{-86}{86} = -1$

Conclusion

Therefore, using the given function rule, we find that $f(-86) = -1$ .

Why is this Important?

Understanding how to apply function rules to solve for specific input values is a crucial skill in mathematics. It allows us to model real-world situations and make predictions based on the relationships between variables.

Real-World Applications

Function rules like the one given in this example can be used to model a wide range of real-world situations, such as:

Finance: A company's revenue function might be modeled using a function rule like $f(x) = \frac{x}{100}$ , where $x$ represents the number of units sold and $f(x)$ represents the total revenue.
Science: A scientist might use a function rule like $f(x) = \frac{x}{10}$ to model the relationship between the amount of a substance and its concentration.
Engineering: An engineer might use a function rule like $f(x) = \frac{x}{50}$ to model the relationship between the amount of a material and its weight.

Tips for Solving Function Rule Problems

When solving function rule problems like this one, here are a few tips to keep in mind:

Read the problem carefully: Make sure you understand what the function rule is and what you are being asked to find.
Substitute the input value: Replace every instance of $x$ with the input value and simplify the resulting expression.
Check your units: Make sure the units of the input value and the output value are consistent.

Common Mistakes to Avoid

When solving function rule problems, here are a few common mistakes to avoid:

Not reading the problem carefully: Make sure you understand what the function rule is and what you are being asked to find.
Not substituting the input value correctly: Make sure to replace every instance of $x$ with the input value and simplify the resulting expression.
Not checking units: Make sure the units of the input value and the output value are consistent.

Conclusion

In conclusion, solving for $f(-86)$ using the given function rule is a straightforward process that involves substituting the input value into the function rule and simplifying the resulting expression. By following the tips and avoiding common mistakes outlined in this article, you can become proficient in solving function rule problems like this one.

Q: What is a function rule?

A: A function rule is a mathematical expression that describes the relationship between an input value and an output value. It is a way to model real-world situations and make predictions based on the relationships between variables.

Q: How do I apply a function rule to solve for a specific input value?

A: To apply a function rule, you need to substitute the input value into the function rule in place of the variable (usually x) and then simplify the resulting expression.

Q: What if the function rule has multiple variables?

A: If the function rule has multiple variables, you need to substitute the input values into the function rule in place of the variables and then simplify the resulting expression.

Q: Can I use a function rule to model real-world situations?

A: Yes, function rules can be used to model a wide range of real-world situations, such as finance, science, and engineering.

Q: How do I check my units when solving a function rule problem?

A: When solving a function rule problem, make sure the units of the input value and the output value are consistent. For example, if the input value is in meters and the output value is in feet, you need to convert the input value to feet before solving the problem.

Q: What if I get a negative result when solving a function rule problem?

A: If you get a negative result when solving a function rule problem, it means that the input value is negative. You can use the absolute value function to find the absolute value of the result.

Q: Can I use a function rule to solve for a specific output value?

A: Yes, you can use a function rule to solve for a specific output value. To do this, you need to rearrange the function rule to isolate the output variable and then substitute the input value into the rearranged function rule.

Q: How do I know if a function rule is linear or nonlinear?

A: A function rule is linear if it can be written in the form f(x) = mx + b, where m and b are constants. A function rule is nonlinear if it cannot be written in this form.

Q: Can I use a function rule to model a situation with multiple variables?

A: Yes, you can use a function rule to model a situation with multiple variables. To do this, you need to substitute the input values into the function rule in place of the variables and then simplify the resulting expression.

Q: How do I graph a function rule?

A: To graph a function rule, you need to plot the input values on the x-axis and the corresponding output values on the y-axis. You can use a graphing calculator or software to help you graph the function rule.

Q: Can I use a function rule to solve a system of equations?

A: Yes, you can use a function rule to solve a system of equations. To do this, you need to substitute the input values into the function rule in place of the variables and then simplify the resulting expression.

Q: How do I know if a function rule is invertible?

A: A function rule is invertible if it has an inverse function. To check if a function rule is invertible, you need to check if the function rule is one-to-one, meaning that each input value corresponds to a unique output value.

Q: Can I use a function rule to model a situation with a constraint?

A: Yes, you can use a function rule to model a situation with a constraint. To do this, you need to substitute the input values into the function rule in place of the variables and then simplify the resulting expression.

Q: How do I know if a function rule is a linear or nonlinear function?

A: A function rule is linear if it can be written in the form f(x) = mx + b, where m and b are constants. A function rule is nonlinear if it cannot be written in this form.

Q: Can I use a function rule to solve a problem with multiple variables and constraints?

A: Yes, you can use a function rule to solve a problem with multiple variables and constraints. To do this, you need to substitute the input values into the function rule in place of the variables and then simplify the resulting expression.

Q: How do I know if a function rule is a function or a relation?

A: A function rule is a function if it assigns each input value to a unique output value. A function rule is a relation if it assigns multiple input values to the same output value.

Q: Can I use a function rule to model a situation with a non-linear relationship?

A: Yes, you can use a function rule to model a situation with a non-linear relationship. To do this, you need to substitute the input values into the function rule in place of the variables and then simplify the resulting expression.

Q: How do I know if a function rule is a quadratic function?

A: A function rule is a quadratic function if it can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants.

Q: Can I use a function rule to solve a problem with a quadratic function?

A: Yes, you can use a function rule to solve a problem with a quadratic function. To do this, you need to substitute the input values into the function rule in place of the variables and then simplify the resulting expression.

Q: How do I know if a function rule is a polynomial function?

A: A function rule is a polynomial function if it can be written in the form f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, and a_0 are constants.

Q: Can I use a function rule to solve a problem with a polynomial function?

A: Yes, you can use a function rule to solve a problem with a polynomial function. To do this, you need to substitute the input values into the function rule in place of the variables and then simplify the resulting expression.

Q: How do I know if a function rule is a rational function?

A: A function rule is a rational function if it can be written in the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomials.

Q: Can I use a function rule to solve a problem with a rational function?

A: Yes, you can use a function rule to solve a problem with a rational function. To do this, you need to substitute the input values into the function rule in place of the variables and then simplify the resulting expression.

Q: How do I know if a function rule is a trigonometric function?

A: A function rule is a trigonometric function if it can be written in the form f(x) = a sin(bx) + c or f(x) = a cos(bx) + c, where a, b, and c are constants.

Q: Can I use a function rule to solve a problem with a trigonometric function?

A: Yes, you can use a function rule to solve a problem with a trigonometric function. To do this, you need to substitute the input values into the function rule in place of the variables and then simplify the resulting expression.

Q: How do I know if a function rule is an exponential function?

A: A function rule is an exponential function if it can be written in the form f(x) = a^x, where a is a constant.

Q: Can I use a function rule to solve a problem with an exponential function?

A: Yes, you can use a function rule to solve a problem with an exponential function. To do this, you need to substitute the input values into the function rule in place of the variables and then simplify the resulting expression.

Q: How do I know if a function rule is a logarithmic function?

A: A function rule is a logarithmic function if it can be written in the form f(x) = log_a(x), where a is a constant.

Q: Can I use a function rule to solve a problem with a logarithmic function?

A: Yes, you can use a function rule to solve a problem with a logarithmic function. To do this, you need to substitute the input values into the function rule in place of the variables and then simplify the resulting expression.

Q: How do I know if a function rule is a piecewise function?

A: A function rule is a piecewise function if it can be written in the form f(x) = {f_1(x) if x < a, f_2(x) if x >= a}, where f_1(x) and f_2(x) are functions and a is a constant.

Q: Can I use a function rule to solve a problem with a piecewise function?

A: Yes, you can use a function rule to solve a problem with a piecewise function. To do this, you need to substitute the input values into the function rule in place of the variables and then simplify the resulting expression.

Q: How do I know if a function rule is a recursive function?

A: A function rule is a recursive function if it can be written in the form f(x) = f(x-1) + g(x), where f(x-1) and g(x) are functions.