Composite Function Numerically, Problem 1: Functions f and g consist of the discrete points in the table, and only these points. Find the values of the composite functions, or explain why no such value exists.

x	f(x)	g(x)
1	3	2
2	5	3
3	4	7
4	2	5
5	1	4

Find g(3) and f(g(3))

Composite Function Numerically, Problem 1: Functions f and g consist of the discrete points in the table, and only these points. Find the values of the composite functions, or explain why no such value exists.

Find g(1) and f(g(1))

Absolute Value Transformations Problem: The previous figure shows the graph of $f(x)=0.5(x-2)^2-4.5$ in the domain $-2 \leq x \leq 6$.

Plot the graph of $y_1=f(x)$. On the same screen, plot $y_2=|f(x)|$ using thick style. Sketch the result and describe how this transformation changes the graph of $f$.

For the following problem,

a. Plot the graph using a window set to show the entire graph, when possible. Sketch the result.

b. Give the y-intercept and any x-intercepts and the locations of any vertical asymptotes.

c. Give the range.

Cubic (polynomial) function $f(x)=x^3-7 x^2+4 x+12$ with the domain $-1 \leq x \leq 7$

Plot the graph of

$$f(x)=0.2 x^2-\frac{|x-3|}{x-3}$$

Use a friendly window that includes x = 3 as a grid point. Sketch the result. Name the feature that appears at x = 3.

Explain why functions with the property f(-x) = -f(x) are called odd functions and functions with the property f(-x) = f(x) are called even functions.

In mathematics you learn things in four ways algebraically, graphically, numerically, and verbally.

In which of previous problems did you go from verbal to graphical? From algebraic to numerical? From numerical to graphical? From graphical to algebraic? From graphical to numerical? From algebraic to graphical?

In mathematics you learn things in four ways algebraically, graphically, numerically, and verbally.

In which of the previous problems was the function given algebraically? Graphically? Numerically? Verbally?

On four copies of y = f(x) in the mentioned figure, sketch the graphs of these four functions: y = -f(x), y = f(-x), y = |f(x)|, and y = f(|x|).

The equation for the function in the previous problem is $f(x)=-3+\sqrt{25-(x-2)^2}$ for $-3 \leq x \leq 7$. Plot the function as $y_1$ on your grapher. Plot $y_2=y_1(|x|)$ using thick style. Does the result confirm your answer to the previous problem, part d?

For the following problem,

a. Plot the graph using a window set to show the entire graph, when possible. Sketch the result.

b. Give the y-intercept and any x-intercepts and the locations of any vertical asymptotes.

c. Give the range.

Quadratic (polynomial) function $f(x)=x^2-6 x+40$ with the domain $0 \leq x \leq 8$

For the following problem, let f(x) = $\sqrt{9-x^2}$.

a. Write the equation for g(x) in terms of x.

b. Plot the graphs of f and g on the same screen. Use a friendly window with integers from about -10 to 10 as grid points. Use the same scale on both axes. Sketch the result.

c. Describe how f(x) was transformed to get g(x), including whether the transformation was an inside or an outside function.

$$g(x)=\frac{1}{2} f(x+3)$$

When you turn on the hot water faucet, the time the water has been running and the temperature of the water are related. Sketch a reasonable graph of this function.

The mentioned figure shows the graph of f(x) = x$^{2}$+ 1 in the domain -1 $\leq$ x $\leq$ 2.

Write an equation for the inverse of the function $y=x^2+1$ by interchanging the variables. Solve the new equation for y in terms of x. How does this solution reveal that there are two different y-values for some x-values?