Math / Algebra

Question42. Find the domain of each function: (A) $f(x)=\frac{2 x-5}{x^{2}-x-6}$ (B) $g(x)=\frac{3 x}{\sqrt{5-x}}$
In Problems 57-59, find the equation of any horizontal asymptote.
57. $f(x)=\frac{5 x+4}{x^{2}-3 x+1}$
58. $f(x)=\frac{3 x^{2}+2 x-1}{4 x^{2}-5 x+3}$
53. Explain how the graph of $m(x)=-|x-4|$ is related to the graph of $y=|x|$ .
54. Explain how the graph of $g(x)=0.3 x^{3}+3$ is related to the graph of $y=x^{3}$ .
19. Complete the square and find the standard form for the quadratic function $f(x)=-x^{2}+4 x$
Then write a brief verbal description of the relationship between the graph of $f$ and the graph of $y=x^{2}$ .

Studdy Solution

STEP 1

1. For finding the domain of a function, we need to identify any restrictions such as division by zero or taking the square root of a negative number.
2. For horizontal asymptotes, we analyze the behavior of the function as $x$ approaches infinity or negative infinity.
3. To complete the square, we need to rewrite the quadratic expression in the form $(x-h)^2 + k$ .
4. Understanding transformations involves recognizing shifts, reflections, and stretches/compressions.

STEP 2

1. Find the domain of function $f(x)$ .
2. Find the domain of function $g(x)$ .
3. Determine the horizontal asymptote for $f(x) = \frac{5x+4}{x^2-3x+1}$ .
4. Determine the horizontal asymptote for $f(x) = \frac{3x^2+2x-1}{4x^2-5x+3}$ .
5. Explain the transformation of $m(x) = -|x-4|$ relative to $y = |x|$ .
6. Explain the transformation of $g(x) = 0.3x^3 + 3$ relative to $y = x^3$ .
7. Complete the square for $f(x) = -x^2 + 4x$ and describe its transformation relative to $y = x^2$ .

STEP 3

Find the domain of $f(x) = \frac{2x-5}{x^2-x-6}$ .
First, identify where the denominator is zero since division by zero is undefined:
$x^2 - x - 6 = 0$
Factor the quadratic:
$(x-3)(x+2) = 0$
The solutions are $x = 3$ and $x = -2$ . Therefore, the domain of $f(x)$ is all real numbers except $x = 3$ and $x = -2$ .

STEP 4

Find the domain of $g(x) = \frac{3x}{\sqrt{5-x}}$ .
The square root in the denominator must be positive, so:
$5 - x > 0$
Solving for $x$ :
$x < 5$
Therefore, the domain of $g(x)$ is all real numbers $x < 5$ .

STEP 5

Determine the horizontal asymptote for $f(x) = \frac{5x+4}{x^2-3x+1}$ .
Since the degree of the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is $y = 0$ .

STEP 6

Determine the horizontal asymptote for $f(x) = \frac{3x^2+2x-1}{4x^2-5x+3}$ .
Since the degrees of the numerator and the denominator are equal, the horizontal asymptote is the ratio of the leading coefficients:
$y = \frac{3}{4}$

STEP 7

Explain the transformation of $m(x) = -|x-4|$ relative to $y = |x|$ .
The graph of $y = |x|$ is shifted 4 units to the right to become $y = |x-4|$ , and then reflected over the x-axis to become $y = -|x-4|$ .

STEP 8

Explain the transformation of $g(x) = 0.3x^3 + 3$ relative to $y = x^3$ .
The graph of $y = x^3$ is vertically compressed by a factor of 0.3 and then shifted up by 3 units.

STEP 9

Complete the square for $f(x) = -x^2 + 4x$ .
First, factor out the negative sign:
$f(x) = -(x^2 - 4x)$
To complete the square, take half of the coefficient of $x$ , square it, and add and subtract it inside the parentheses:
$f(x) = -(x^2 - 4x + 4 - 4)$ $f(x) = -((x-2)^2 - 4)$ $f(x) = -(x-2)^2 + 4$
The standard form is $f(x) = -(x-2)^2 + 4$ .
Describe its transformation relative to $y = x^2$ :
The graph of $y = x^2$ is reflected over the x-axis, shifted 2 units to the right, and 4 units up.

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Studdy Solution

STEP 1

STEP 2

STEP 3

STEP 4

Find the domain of $g(x) = \frac{3x}{\sqrt{5-x}}$ .
The square root in the denominator must be positive, so:
$5 - x > 0$
Solving for $x$ :
$x < 5$
Therefore, the domain of $g(x)$ is all real numbers $x < 5$ .

STEP 5

Determine the horizontal asymptote for $f(x) = \frac{5x+4}{x^2-3x+1}$ .
Since the degree of the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is $y = 0$ .

STEP 6

Determine the horizontal asymptote for $f(x) = \frac{3x^2+2x-1}{4x^2-5x+3}$ .
Since the degrees of the numerator and the denominator are equal, the horizontal asymptote is the ratio of the leading coefficients:
$y = \frac{3}{4}$

STEP 7

Explain the transformation of $m(x) = -|x-4|$ relative to $y = |x|$ .
The graph of $y = |x|$ is shifted 4 units to the right to become $y = |x-4|$ , and then reflected over the x-axis to become $y = -|x-4|$ .

STEP 8

Explain the transformation of $g(x) = 0.3x^3 + 3$ relative to $y = x^3$ .
The graph of $y = x^3$ is vertically compressed by a factor of 0.3 and then shifted up by 3 units.

STEP 9

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Studdy Solution

STEP 1

STEP 2

STEP 3

STEP 4

Find the domain of g(x)=3x5−x g(x) = \frac{3x}{\sqrt{5-x}} g(x)=5−x​3x​.The square root in the denominator must be positive, so:5−x>0 5 - x > 0 5−x>0Solving for x x x:x<5 x < 5 x<5Therefore, the domain of g(x) g(x) g(x) is all real numbers x<5 x < 5 x<5.

STEP 5

Determine the horizontal asymptote for f(x)=5x+4x2−3x+1 f(x) = \frac{5x+4}{x^2-3x+1} f(x)=x2−3x+15x+4​.Since the degree of the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is y=0 y = 0 y=0.

STEP 6

Determine the horizontal asymptote for f(x)=3x2+2x−14x2−5x+3 f(x) = \frac{3x^2+2x-1}{4x^2-5x+3} f(x)=4x2−5x+33x2+2x−1​.Since the degrees of the numerator and the denominator are equal, the horizontal asymptote is the ratio of the leading coefficients:y=34 y = \frac{3}{4} y=43​

STEP 7

STEP 8

Explain the transformation of g(x)=0.3x3+3 g(x) = 0.3x^3 + 3 g(x)=0.3x3+3 relative to y=x3 y = x^3 y=x3.The graph of y=x3 y = x^3 y=x3 is vertically compressed by a factor of 0.3 and then shifted up by 3 units.

STEP 9

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Find the domain of $g(x) = \frac{3x}{\sqrt{5-x}}$ .
The square root in the denominator must be positive, so:
$5 - x > 0$
Solving for $x$ :
$x < 5$
Therefore, the domain of $g(x)$ is all real numbers $x < 5$ .

Determine the horizontal asymptote for $f(x) = \frac{5x+4}{x^2-3x+1}$ .
Since the degree of the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is $y = 0$ .

Determine the horizontal asymptote for $f(x) = \frac{3x^2+2x-1}{4x^2-5x+3}$ .
Since the degrees of the numerator and the denominator are equal, the horizontal asymptote is the ratio of the leading coefficients:
$y = \frac{3}{4}$

Explain the transformation of $g(x) = 0.3x^3 + 3$ relative to $y = x^3$ .
The graph of $y = x^3$ is vertically compressed by a factor of 0.3 and then shifted up by 3 units.