How To Add And Multiply Two Piecewise Functions

Scalar Multiplication

To multiply a function past a scalar, multiply each output by that scalar. For instance, if f (x) = 4x - 1, then f (x) = (4x - one) = iix - . If g(ten) = x - 2, then threeg(x) = three(x - 2) = 310 - 6. If h(x) = x ² + ii, so -2h(10) = - 2(x ² +ii) = - 2ten ² - 4. (3x)h(x) = (three10)(x ^two +2) = threex ⁱⁱⁱ + 6x .

The y -coordinate of each betoken on the graph of f (x) is the result of multiplying the y -coordinate of f by .

Multiplication of Functions

To multiply a role by another office, multiply their outputs. For example, if f (x) = 2x and g(x) = ten + 1, then fg(3) = f (3)×g(3) = 6×iv = 24. fg(x) = 2x(ten + one) = 2x ² + 10 .

Compound Functions

When nosotros take f (g(x)), we take g(x) as the input of the function f . In other words, we take 10 as the input of chiliad and evaluate g(x), and then we take this issue every bit the input of f and evaluate f (g(x)).

For example, if f (10) = 10x and yard(x) = 10 + 1, then to find f (g(4)), we find g(4) = four + one + v, and and then evaluate f (5) = 10(five) = 50. Similarly, f (g(12)) = f (12 + ane) = f (13) = 10(xiii) = 130. In general, f (g(x)) = f (10 + 1) = 10(x + i) = x10 + 10.

Case: f (x) = 2x - 2, g(x) = x ^two - 8.

f (g(iii)) = f (three² - 8) = f (i) = 0.
f (g(- 4)) = f ((- 4)ⁱⁱ - 8) = f (eight) = 2(eight) - 2 = xiv.

In general, f (g(x)) = f (x ² -8) = ii(ten ^two -8) - 2 = 2ten ^two - 18.
m(f (iii)) = g(2(3) - two) = thousand(4) = 4² - 8 = eight.
thou(f (- 4)) = g(2(- 4) - 2) = thou(- 10) = (- 10)² - 8 = 92.

In general, g(f (x)) = yard(2x - ii) = (2x - ii)² -8 = 4ten ^two - eightx - iv.

f (g(ten)) is denoted fochiliad(x) and m(f (10)) is denoted gof (10). Note that it is not necessarily true that fok(ten) = thousandof (10), as shown in the above example.