Solving a Function for a Variable
In our previous lesson, we explored the process of evaluating a function when provided with an input value. Here, we'll explore this topic more and learn how to solve a function for a variable. Solving aims to find the corresponding input value when the output is given.
Just as in our earlier lesson, we can solve a function algebraically, graphical analysis, or a table. For this lesson, we will use rational numbers as our output values.
How to Solve Algebraically
When solving a function algebraically, we're presented with the output value, often shown as y or f(x). It's crucial to remember the relationship y = f(x).
To solve a function algebraically, we have to substitute in the given y-value, or f(x), and then solve for x using techniques learned when isolating a variable. In this lesson, we will focus on solving linear functions, but different techniques need to be used for other functions, like quadratics.
If you need a refresher on the algebraic concepts required, and the properties of operations check out our resource here (coming soon!).
Let's work through an example problem.
Example 1.
For the function f(x) = 4x + 7, if f(x) = 55, solve for x.
f(X)= 4x + 7
Substitute 55 for f(x)
55 = 4x + 7
2. Subtract 7 from both sides
55 = 4x + 7
-7 -7
48 = 4x
3. Divide both sides by 4
48 = 4x
4 4
12 = 1x
4. Interpret the answer: When f(x) = 55, x = 12
When we solve algebraically, we have to “undo” the problem and isolate the variable by using inverse operation and other property of operations.
Let’s look at one more example.
Example 2.
For the function f(x) = (-1/2) x – 1, if f(x) = 3, solve for x.
f(x) = (-1/2) x – 1
Substitute 3 for f(x)
3 = (-1/2) x – 1
2. Add 1 to both sides
3 = (-1/2) x – 1
+1 +1
4 = (-1/2) x
3. Multiply by the reciprocal of (-1/2)
(-2/1) • 4 = (-1/2) x • (-2/1)
-8 = x
4. Interpret the answer: When f(x) = 3, x = -8
Another way to view solving a function algebraically is finding the (x, y) coordinate pair. It's important to note that one x-value corresponds to exactly one y-value for a function. However, one y-value can correspond to multiple x-values. Attention to detail, especially in cases where the function exhibits symmetry, is essential to identify all possible inputs (because there may be more than one).
While algebraically, the example focused on solving linear functions, which will only result in one solution, but for graphs and tables, we can solve for many x values based on the type of function.
Graphical Evaluation
Graphical representations provide another avenue for solving a function. By inspecting a function graph, we can pinpoint the x-value, or values, that correspond to a given y-value, presented as an (x, y) coordinate pair.
Let’s take a look at our previous example in graph form.
Example 3.
Two graphs showing the function f(x) = 4x + 7. The first graph shows the linear function plotted on a coordinate plane. The second graph highlights solving the function for f(x) = 55, showing the intersection point where x = 12. A brown dashed line from y = 55 intersects the function at the point (12, 55).
When f(x) or y is equal to 55, the x value is equal to 12. The coordinate point would be (12, 55).
Now, let’s look at another type of function and solve it.
The graph below is the function
f(x) = (x + 3)(x - 1)(x - 5),
solve for x when f(x) = 0.
Example 4.
Two graphs displaying the function f(x) = (x + 3)(x - 1)(x - 5). The first graph shows the full curve of the function on a coordinate plane. The second graph highlights the points where the function equals zero, with brown markers at x = -3, x = 1, and x = 5. A note below indicates these x-values when f(x) = 0.
To solve this, we look for where the y-coordinate on the graph is equal to 0. Then, we find the corresponding x-coordinate (s). In this problem, multiple x-values correspond to when y = 0.
When x = -3, 1, and 5, f(x) = 0.
Keep in mind that the objective of solving is to find the x-value(s) for a given y-value.
Table-Based Evaluation
Alternatively, a table can be used to solve a function. The structured view of the function in a table makes locating the corresponding x-value for a given y-value or f(x) is straightforward.
Let’s take a look at the table given below and solve for when f(x)=15
Example 5.
Two tables showing x and y values. The first table lists pairs: (0, 7), (1, 11), (2, 15), and (3, 19). An arrow points to the second table, where the pair (2, 15) is circled, indicating that when f(x) = 15, x = 2.
In the x,y table, we look for when the y-value equals 15. In the example, 15 only appears once, and the corresponding x-value is 2.
So, when f(x) = 15, x = 2.
Let’s look at another table example. Given the table below, solve for when f(x) = 0.
Example 6.
Two tables showing x and y values. The first table lists pairs: (-2,21), (-1,24), (0,15), and (1,0). An arrow points to the second table, where the pair (1,0) is circled, indicating that when f(x) = 0, x = 1.
In the example, 0 also only appears once, and the corresponding x-value is 1.
So, when f(x) = 0, x = 1.
It is possible that the f(x) value could appear more than once, and there could be more than one x-value that corresponds to the given y-value.
Practice Time!
Now, let's test our skills. Try solving the following function problems. The answers are at the end of this post.
Three problems labeled A-C with directions stating to solve for the value of x, when given the f(x). Problem A reads, If f(x)=(-1/2)x-1 and f(x)=9, B reads if f(x)=-4 and gives the graph f(x) = -absval(x-3)+7, and C reads f(x)=6 and a table with the coordinates (-2,2), (0,4), (2,6), and (4,6).
Conclusion
In the process of solving functions, we are provided with a y-value (output) and are tasked with determining the corresponding x-value (input). Depending on how the problem is given, solving can be achieved through algebraic methods, graphical analysis, or tables. Mastery across all three representations enhances our proficiency in solving problems.
In our upcoming lesson, we'll apply everything we've learned in our Introduction to Function Series and explore Function Applications and Word Problems.
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x = 16
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x = -8, x = 14
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x = 2, x = 4
