Revisiting Function Equality

Let us revisit function equality. Consider the simple scenario described below.

• If f(x) = 5x and g(x) = ax, and if f(x) = g(x) for all values of x, then what is the value of a?

This may seem like a trivial example, but it applies to problems of the same general type. So, let us look at the following table.

x	f(x)	g(x)	a
-5	-25	-5a	5
-4	-20	-4a	5
-3	-15	-3a	5
-2	-10	-2a	5
-1	-5	-a	5
0	0	0	anything
1	5	a	5
2	10	2a	5
3	15	3a	5
4	20	4a	5
5	25	5a	5

Only when x = 0 will a take on any value other than 5, but since the two functions must be equal for all values of x, a must be 5. This idea can be expanded to more complicated problems. Let us look at another example.

• If (a + 7)(x² + 3x - 4) = (2x + b)(2x + 8) for all values of x, then what is the value of b?

There are at least two ways to solve this problem. One of them, the one that you are likely to default to, will take much longer than the other. The longer, brute force method has you applying the Distributive Property to both sides and getting the following.

• ax² + 7x² + 3ax + 21x - 4a - 28 = 4x² + 16x + 2bx + 8b

Now, we can collect like terms.

• (a + 7)x² + (3a + 21)x - 4(a + 7) = 4x² + (2b + 16)x + 8b

From the previous example, it should be evident that these two expressions will only be equal if the corresponding coefficients are equal.

• (a + 7)x² = 4x² → a + 7 = 4
• (3a + 21)x = (2b + 16)x → 3a + 21 = 2b + 16
• -4(a + 7) = 8b

From the first of these three equations, we can see that a = -3. We can then use this value in either of the other equations to conclude that b = -2. However, this can be solved with far less work by analyzing the equation more closely early on. Let us rewrite the problem.

• If (a + 7)(x² + 3x - 4) = (2x + b)(2x + 8) for all values of x, then what is the value of b?

If we look at the right side of the equation, we can clearly see that the x² term will be 4x². This then forces the x² term on the left to also be 4x², forcing a + 7 = 4. If a + 7 = 4, then the constant term on the left hand side will have to equal -16, forcing the constant term on the right hand side, 8b, to also equal -16. If 8b = -16, b = -2.

We get to the same answer with either method, but if you practice the second of the two methods, you can become more proficient at getting to the answer faster.