### Chapter 5 - Algebra Coach Exercises

**Algebra Coach Exercise on substituting
expressions into functions.**

Each question contains two formulas.
The second formula defines a function called *f* with argument *x*.
The first formula gives an expression to substitute in for *x*.
Drag-and-drop each of these problems into the
Algebra Coach and click the Substitute button to carry out the substitution.
- x = 5, f (x) = x^3 + 10 x +2
- x = a, f (x) = sqrt (5 - x^2)
- x = x + 2, f (x) = sqrt (5 - x^2)
- x = x + h, f (x) = 1 / sqrt (5 - x^2)

**Algebra Coach Exercise on composition
of functions.**

Each question below contains two formulas.
The first formula defines a function *y = f* (*x*) and
the second formula defines a function *z = g* (*y*).
Substituting the first function, *f*, into the second
function, *g*, produces the composite function
, like this:
To do this in the Algebra Coach drag-and-drop
each of these problems into the Algebra Coach and click the Substitute button.
- y = x + 3, z = y^2 - 2 y
- y = x^2 - 2 x, z = y + 3
- y = x + 2, z = sqrt (5 - y^2)
- y = x + h, z = 1 / sqrt (5 - y^2)

**Algebra Coach Exercise on the inverse
of a function.**

Each question below contains two formulas.
The first formula defines a function *y = f* (*x*) and
the second formula defines a function *z = g* (*y*).
Substituting the first function, *f*, into the second
function, *g*, produces the composite function
, like this:
We claim that *g* is the inverse function of *f*.
If this is true then the composite function should have no overall
effect and we should find that *z* = *x* (the output
should equal the input).
To check this claim in the Algebra Coach drag-and-drop
each of these problems into the Algebra Coach and click the Substitute button.
Then click on the *Do to right side* button and then on the Simplify
button repeatedly.
- y = 3 x, z = y / 3
- y = 2 x + 3, z = (y - 3) / 2
- y = -2 x + 3, z = (-y + 3) / 2
- y = 5 / 9 (x - 32), z = 9 y / 5 + 32
- y = (arcsin(x) - 4)/3, z = sin(3y + 4)
- y = log(x/3), z = 3 * 10^y