Functions and Graphs

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Algebra 2 › Functions and Graphs

Questions 1 - 10

If and , what is $g(f(x))\cdot f(x)$ ?

$\textup{The answer is not given.}$

Explanation

Evaluate first. Substitute the function into .

Distribute the integer through the binomial and simplify the equation.

Multiply this expression with .

The answer is:

What is the domain of the following function? Please use interval notation.

$y=\left | -x\right |$

$(-\infty ,\infty )$

$(0,\infty )$

$(-\infty ,0)$

Explanation

A basic knowledge of absolute value and its functions is valuable for this problem. However, if you do not know what the typical shape of an absoluate value function looks like, one can always plug in values and plot points.

Upon doing so, we learn that the -values (domain) are not restricted on either end of the function, creating a domain of negative infinity to postive infinity.

If we plug in -100000 for , we get 100000 for .

If we plug in 100000 for , we get 100000 for .

Additionally, if we plug in any value for , we will see that we always get a real, defined value for .

**Extra Note: Due to the absolute value notation, the negative (-) next to the is not important, in that it will always be made positive by the absolute value, making this function the same as $y=\left | x\right |$ . If the negative (-) was outside of the absolute value, this would flip the function, making all corresponding -values negative. However, this knowledge is most important for range, rather than domain.

Find the range of the function:

$(-\infty, 4]$

$(-\infty, 4)$

$(-\infty, -4]$

$(-\infty, 4)\bigcup(4,\infty)$

$(-\infty,\infty )$

Explanation

The range is the existing y-values that contains the function.

Notice that this is a parabola that opens downward, and the y-intercept is four.

This means that the highest y-value on this graph is four. The y-values will approach negative infinity as the domain, or x-values, approaches to positive and negative infinity.

$x\leq4$

The answer is: $(-\infty, 4]$

Determine the inverse of: $y=\frac{2}{5}x-\frac{1}{2}$

$y=\frac{5}{2}x+\frac{5}{4}$

$y=\frac{5}{2}x+\frac{1}{2}$

$y=-\frac{5}{2}x-\frac{1}{2}$

$y=-\frac{5}{2}x+\frac{1}{2}$

$y=\frac{5}{2}x-\frac{5}{4}$

Explanation

Interchange the x and y-variables.

$x=\frac{2}{5}y-\frac{1}{2}$

Solve for y. Add one-half on both sides.

$x+\frac{1}{2}=\frac{2}{5}y-\frac{1}{2}+\frac{1}{2}$

Simplify both sides.

$x+\frac{1}{2}=\frac{2}{5}y$

Multiply five over two on both sides in order to isolate the y-variable.

$\frac{5}{2}(x+\frac{1}{2})=\frac{2}{5}y\cdot \frac{5}{2}$

Apply the distributive property on the left side. The right side will reduce to just a lone y-variable.

The answer is: $y=\frac{5}{2}x+\frac{5}{4}$

Based on the figure below, which line depicts a quadratic function?

Red line

Blue line

Green line

Purple line

None of them

Explanation

A parabola is one example of a quadratic function, regardless of whether it points upwards or downwards.

The red line represents a quadratic function and will have a formula similar to $\small {f(x)=ax^2+bx+c}$ .

The blue line represents a linear function and will have a formula similar to $\small {f(x)=ax+b$ .

The green line represents an exponential function and will have a formula similar to $\small {f(x)=ax^b}$ .

The purple line represents an absolute value function and will have a formula similar to $\small {f(x)=\left |ax+b \right |}.$ .

Consider the following two functions:

$f(x) = x^{2}$ and $g(x) = (x+3)^{2} - 6$

How is the function shifted compared with ?

units left, units down

units right, units down

units left, units up

units right, units down

units left, units down

Explanation

The $(x+3)^{2}$ portion results in the graph being shifted 3 units to the left, while the results in the graph being shifted six units down. Vertical shifts are the same sign as the number outside the parentheses, while horizontal shifts are the OPPOSITE direction as the sign inside the parentheses, associated with .

$\textup{What is the domain of y = x ?}$

$x\neq0$

Explanation

All inputs are valid. There is nothing you can put in for x that won't work.

What are the -intercepts of the equation?

$y=\frac{x^2-9}{x^2-16}$

There are no -intercepts.

Explanation

To find the x-intercepts of the equation, we set the numerator equal to zero.

$\sqrt{9}=\sqrt{x^2}$

Find the range of the function:

$(-\infty, 4]$

$(-\infty, 4)$

$(-\infty, -4]$

$(-\infty, 4)\bigcup(4,\infty)$

$(-\infty,\infty )$

Explanation

The range is the existing y-values that contains the function.

Notice that this is a parabola that opens downward, and the y-intercept is four.

This means that the highest y-value on this graph is four. The y-values will approach negative infinity as the domain, or x-values, approaches to positive and negative infinity.

$x\leq4$

The answer is: $(-\infty, 4]$

Determine the inverse: