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Functions for reference:
- [tex]\( f(x) = x + 5 \)[/tex]
- [tex]\( g(x) = x^2 + 6x + 5 \)[/tex]
- [tex]\( h(x) = 2x - 3 \)[/tex]
- [tex]\( j(x) = -4x \)[/tex]

Using the same functions given above, find [tex]\((h - f)(x)\)[/tex].

1. Write the functions, subtracting in the order given.
[tex]\[
(h - f)(x) = 2x - 3 - (x + 5)
\][/tex]

2. Distribute the subtraction to each term.
[tex]\[
(h - f)(x) = 2x - 3 - x - 5
\][/tex]

3. Rearrange to prepare to combine like terms.
[tex]\[
(h - f)(x) = 2x - x - 3 - 5
\][/tex]

4. Combine like terms and ensure the new function is in standard form.
[tex]\[
(h - f)(x) = x - 8
\][/tex]

Now, let's try another. Find [tex]\( m(x) = (g - j)(x) \)[/tex].

1. Write the functions, subtracting in the order given.
[tex]\[
m(x) = x^2 + 6x + 5 - (-4x)
\][/tex]

2. Distribute the subtraction to each term. (Hint: Subtracting a negative is the same as adding a positive. We'll rewrite with this in mind.)
[tex]\[
m(x) = x^2 + 6x + 5 + 4x
\][/tex]

3. Combine like terms and ensure the new function is in standard form.
[tex]\[
m(x) = x^2 + 10x + 5
\][/tex]

Sagot :

GinnyAnsweridn
Let's start by finding [tex]\( (h - f)(x) \)[/tex]:

1. Write the functions, subtracting them in the order given:
[tex]\[ (h - f)(x) = h(x) - f(x) = (2x - 3) - (x + 5) \][/tex]

2. Distribute the subtraction to each term:
[tex]\[ (h - f)(x) = 2x - 3 - x - 5 \][/tex]

3. Rearrange to prepare to combine like terms:
[tex]\[ (h - f)(x) = 2x - x - 3 - 5 \][/tex]

4. Combine like terms:
[tex]\[ (h - f)(x) = (2x - x) - (3 + 5) = x - 8 \][/tex]

So, [tex]\( (h - f)(x) = x - 8 \)[/tex].

Now let's find [tex]\( m(x) = (g - j)(x) \)[/tex]:

1. Write the functions, subtracting them in the order given:
[tex]\[ m(x) = g(x) - j(x) = (x^2 + 6x + 5) - (-4x) \][/tex]

2. Distribute the subtraction to each term (subtracting a negative is the same as adding a positive):
[tex]\[ m(x) = x^2 + 6x + 5 + 4x \][/tex]

3. Combine like terms and make sure the new function is in standard form:
[tex]\[ m(x) = x^2 + (6x + 4x) + 5 = x^2 + 10x + 5 \][/tex]

So, [tex]\( m(x) = x^2 + 10x + 5 \)[/tex].

Summarizing the results:
- [tex]\( (h - f)(x) = x - 8 \)[/tex]
- [tex]\( m(x) = (g - j)(x) = x^2 + 10x + 5 \)[/tex]
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