Q: Given $$f(x) = 3x - 2$$ and $$g(x) = \sqrt{x}$$, find $$f \circ g(x)$$ and its domain.

$$3\sqrt{x} - 2$$; \text{Domain: } [0, \infty)$

Q: Given $$f(x) = 71x - 2$$ and $$g(x) = x + 10$$, find $$fg(x)$$ and its domain.

$$71x - 2$$; \text{Domain: } (-\infty, \infty)$

Question 1

Solve the equation $$y^2 - 8y - 3 = 0$ by completing the square.

Accepted Answer

$$y = 4 \pm \sqrt{19}$$

Question 2

For the line segment bounded by the endpoints $$(3, 9)$$ and $$(13, 15)$$, find the midpoint.

Accepted Answer

$$(8, 12)$$

Question 3

Given $$f(x) = 3x - 2$$ and $$g(x) = \sqrt{x}$$, find $$f \circ g(x)$$ and its domain.

Accepted Answer

$$3\sqrt{x} - 2$$; 	ext{Domain: } [0, \infty)$

Question 4

Which graph represents the transformation of $$f(x) = |x|$$ into $$g(x) = 2|4x + 3| + 3$$?

Accepted Answer

Image 3

Question 5

Graph $$h(x) = \sqrt{2x - 6} as a $$transformation of the square root function $$f(x) = \sqrt{2x}. $$Which image matches $$h(x)$$?

Accepted Answer

Image 7

Question 6

Given $$f(x) = 71x - 2$$ and $$g(x) = x + 10$$, find $$fg(x)$$ and its domain.

Accepted Answer

$$71x - 2$$; 	ext{Domain: } (-\infty, \infty)$

Question 7

For the one-to-one function $$f(x) = \frac{x}{2}$$, find the inverse function $$f^{-1}(x$$)$$.

Accepted Answer

$$f^{-1}(x) = 2x$

Question 8

Find $$(f \circ g$$)^{-1}$$(x)$$ and $$(g^{-1}$$ \circ $$f^{-1})(x$$)$$ if f(x) = 5x$ and g(x) = x - 7$.

Accepted Answer

$$(f \circ g$$)^{-1}$$(x) = \frac{x + 35}{5},\ (g$$^{-1}$$ \circ f$$^{-1}$$)(x) = \frac{x}{5} + 7$$

Question 9

A medical imaging device uses the function $$f(x) = |x| to $$process signals. If the device is recalibrated so that the new function is $$g(x) = 2|4x + 3| + 3$$, what is the vertex of the new function?

Accepted Answer

$$(-0.75, 3)$$

Question 10

A chemist models the concentration of a substance with $$h(x) = \sqrt{2x - 6}. $$What is the minimum value of $$x$$ for which the concentration is defined?

Accepted Answer

$$x = 3$$

Question 11

Which of the following graphs best represents the function $$g(x) = \sqrt{25x}$$?

Accepted Answer

Image 13

Question 12

A population growth model uses the function $$f(x) = \sqrt{x}$$, but due to a shift in the environment, the model changes to $$h(x) = \sqrt{x} + 3. $$Which graph represents $$h(x)$$?

Accepted Answer

Image 7

Question 13

A physicist is analyzing the function $$f(x) = \sqrt{x}$$ and its transformation $$h(x) = \sqrt{x - 3}. $$Which graph corresponds to $$h(x)$$?

Accepted Answer

Image 9

Question 14

A financial analyst uses the function $$f(x) = \sqrt{x} to $$model returns. If the model is shifted left by 3 units, which graph represents the new function $$h(x) = \sqrt{x + 3}$$?

Accepted Answer

Image 8

Question 15

A biologist uses the function $$f(x) = \sqrt{x} to $$model cell growth. If the model is reflected over the $$x-$$axis and shifted down by 3 units, which graph represents $$h(x) = -\sqrt{x} - 3$$?

Accepted Answer

Image 6

Precalculus Study Guide: Analysis of Graphs of Functions

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