How to Solve Basic Functions Under Binary Operations; Solving f(x)
Not everyone loves Mathematics. Many students wonder why it’s necessary to learn in the first place.
Well, Mathematics is unique and requires one to think systematically. It is everywhere. It helps to solve problems in life and this is unknown to some people.
For example, how can we let robots work without our knowledge of Mathematics?
How does a computer operate? This requires humans to use mathematics to build on the device.
Let’s be open-minded and optimistic towards learning Mathematics.
The expression f(x) means “The function of x” or “f of x”. f(x) is pronounced as F of x.
“f(x)” is a different way of writing “y” in equations. “f(x)” and “y” means the same thing but “f(x)” allows flexibility.
Meanwhile, it doesn’t have to be “f(x)”. The function name could be g(x), m(x), v(x), etc, except “x(x)” as this could make your work confusing.
Function notation allows us to use more than one function at a time and this is to prevent the use of one function that can be confusing in a single context.
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f(x) equates to an output, each input has a single output. For any function f(x), x is called the input or the argument of the function.
To evaluate a function, all you need to do is change the argument of the function from x to the new given argument.
You do this by substituting the value into the input variable of the function and simplify.
For Example: f(x)= 4x (read as “f of x equals 4x” or “the value of the function at x is 4x”) is an example of function notation.
“f” is the function name, “x” is the function’s input i.e. the value put into the function and “4x” is the function’s output.
It is a function because each input “x” has a single output “4x”:
f(2) = 4(2)
f(3) = 4(3)
f(10) = 4(10)
Do not pronounce “f(x)” as “f times x” or try to multiply the function with the input.
Now, let’s say “f(x) = 4x + 6, find f(-2)” (pronounced as f of x equals 4x plus 6; find f-of-negative2).
To solve this, you plug -2 for x, multiply by 4 and add 6. The final value would be -2.
f(x) = 4x + 6
f(-2) = 4(-2) + 6
= -8 + 6
Another example: “Given f(x) = x² find f(4)” (read as f of x equals x raised to power 2).
To solve this equation, substitute 4 by x and multiply 4 by 4.
f(x) = x²
f(4) = 4²
= 4 × 4
Therefore, the final value is 16.
Example 3: g(x) = 2x + 9x — 7, find g(x+2) [Pronounced as g of x equals to 2x plus 9x minus 7]
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Firstly, substitute x by (x + 2) in the formula of function g
g(x+2) = 2(x+2) + 9(x+2) — 7
Expand the equation
g(x+2) = 2x+4 +9x +18 -7
Collect like terms
= 2x + 9x + 18 + 4 – 7
= 11x + 22 – 7
= 11x + 15.
Let’s examine this:
Given f(x) = 3×² -7x -5, find f(x-2) [read as f of x equals 3x raised to power 2 minus 7x minus 5].
Substitute x by x-2
f(x-2) = 3(x-2)² — 7(x-2) -5
Expand the terms
f(x-2) = 3(x²-4x+4) — 7x+14 – 5
= 3×² — 12x + 12 — 7x + 14 – 5
Collect like terms
= 3×² — 12x — 7x + 14 + 12 – 5
= 3×² -19x + 21.
Another Example: f(x) = x² + 3x — 1, find f(-2)
To evaluate, substitute -2 by x.
f(-2) = -2² + 3(-2) — 1
f(-2) = 4 – 6 – 1
Then, my final answer is:
f(-2) = -3
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If you want to avoid mistakes when working with negatives, make use of parentheses () as I did above.
It helps to keep track of things like whether the power is on the minus(-) sign.
In conclusion, solve various equations to improve your knowledge and get accurate answers
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